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Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 950134, 12 pages http://dx.doi.org/10.1155/2013/950134 Research Article Partial Regularity for Nonlinear Subelliptic Systems with Dini Continuous Coefficients in Heisenberg Groups Jialin Wang, Pingzhou Hong, Dongni Liao, and Zefeng Yu School of Mathematics and Computer Science, Gannan Normal University, Ganzhou, Jiangxi 341000, China Correspondence should be addressed to Jialin Wang; [email protected] Received 17 June 2013; Accepted 8 August 2013 Academic Editor: Pekka Koskela Copyright © 2013 Jialin Wang et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. is paper is concerned with partial regularity to nonlinear subelliptic systems with Dini continuous coefficients under quadratic controllable growth conditions in the Heisenberg group H . Based on a generalization of the technique of A-harmonic approximation introduced by Duzaar and Steffen, partial regularity to the sub-elliptic system is established in the Heisenberg group. Our result is optimal in the sense that in the case of H¨ older continuous coefficients we establish the optimal H¨ older exponent for the horizontal gradients of the weak solution on its regular set. 1. Introduction and Statements of Main Results In this paper, we are concerned with partial regularity of weak solutions to nonlinear sub-elliptic systems of equations of second order in the Heisenberg group H in divergence form, and more precisely, we consider the following systems: − 2 ∑ =1 (, () , ()) = (, () , ()) in Ω, (1) where Ω is a bounded domain in H , = { 1 ,..., 2 }, the definition of ( = 1, . . . , 2) is to be seen in the next section (11), = ( 1 ,..., ):Ω→ R , (, , ) : R 2+1 × R × R 2 → R 2 , and (, , ) : R 2+1 × R × R 2 → R . Under the coefficients assumed to be Dini continuous, the aim of this paper is to establish optimal partial regularity to the sub-elliptic system (1) in the Heisenberg group H . Comparing H¨ older continuous coefficients (see [1, 2] for the case of sub-elliptic systems), such assumption is weaker. More precisely, we assume for the continuity of with respect to the variables (, ) that (1 + ) −1 (, , ) − ( , , ) ≤ (||) ( (, ) + | − |) (2) for all , ∈ Ω, , ∈ R , and ∈ R 2 , where : (0, +∞) → [1, +∞) is monotone nondecreasing and : (0, +∞) → [0, +∞) is monotone nondecreasing and concave with (0+) = 0. We also required that → − () be nonincreasing for some ∈ (0, 1) and that () = ∫ 0 () < ∞ for some > 0. (3) We adopt the method of A-harmonic approximation to the case of sub-elliptic systems in the Heisenberg groups and establish the optimal partial regularity result. Roughly speaking, assume additionally to the standard hypotheses (see precisely (H1), (H2), and (H4)) that (1 + ||) −1 (, , ) satisfies (2) and (3). Let ∈ 1,2 (Ω, R ) be a weak solution of the sub-elliptic system (1). en, is of class 1 outside a closed singular set Sing ⊂ Ω of the Haar measure 0. Furthermore, for 0 ∈Ω\ Sing, the derivative of has the modulus of continuity → () in a neighborhood of 0 . Our result is optimal in the sense that in the case () = , 0<<1, we have () = −1 H¨ older continuity Γ 1, to be optimal in that case. As is well known, even under reasonable assumptions on and of the systems of equations, one cannot, in general, expect that weak solutions of (1) will be classical, that is, 2 -solutions. is was first shown by de Giorgi [3];
Transcript
Page 1: Research Article Partial Regularity for Nonlinear Subelliptic Systems …downloads.hindawi.com/journals/aaa/2013/950134.pdf · 2019-07-31 · Partial Regularity for Nonlinear Subelliptic

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 950134 12 pageshttpdxdoiorg1011552013950134

Research ArticlePartial Regularity for Nonlinear Subelliptic Systems with DiniContinuous Coefficients in Heisenberg Groups

Jialin Wang Pingzhou Hong Dongni Liao and Zefeng Yu

School of Mathematics and Computer Science Gannan Normal University Ganzhou Jiangxi 341000 China

Correspondence should be addressed to Jialin Wang jialinwang1025hotmailcom

Received 17 June 2013 Accepted 8 August 2013

Academic Editor Pekka Koskela

Copyright copy 2013 Jialin Wang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper is concerned with partial regularity to nonlinear subelliptic systems with Dini continuous coefficients under quadraticcontrollable growth conditions in the Heisenberg group H119899 Based on a generalization of the technique of A-harmonicapproximation introduced byDuzaar and Steffen partial regularity to the sub-elliptic system is established in theHeisenberg groupOur result is optimal in the sense that in the case of Holder continuous coefficients we establish the optimal Holder exponent forthe horizontal gradients of the weak solution on its regular set

1 Introduction and Statements ofMain Results

In this paper we are concernedwith partial regularity of weaksolutions to nonlinear sub-elliptic systems of equations ofsecond order in the Heisenberg groupH119899 in divergence formand more precisely we consider the following systems

minus

2119899

sum

119894=1

119883

119894119860

120572

119894(120585 119906 (120585) 119883119906 (120585)) = 119861

120572

(120585 119906 (120585) 119883119906 (120585)) in Ω

(1)

where Ω is a bounded domain in H119899 119883 = 119883

1 119883

2119899 the

definition of119883119894(119894 = 1 2119899) is to be seen in the next section

(11) 119906 = (1199061 119906119873) Ω rarr R119873 119860120572119894(120585 119906 119901) R2119899+1 times R119873 times

R2119899119873 rarr R2119899119873 and 119861120572(120585 119906 119901) R2119899+1 timesR119873 timesR2119899119873 rarr R119873Under the coefficients119860120572

119894assumed to beDini continuous

the aim of this paper is to establish optimal partial regularityto the sub-elliptic system (1) in the Heisenberg group H119899Comparing Holder continuous coefficients (see [1 2] for thecase of sub-elliptic systems) such assumption is weakerMoreprecisely we assume for the continuity of 119860120572

119894with respect to

the variables (120585 119906) that

(1 +

1003816

1003816

1003816

1003816

119901

1003816

1003816

1003816

1003816

)

minus1 10038161003816

1003816

1003816

1003816

119860

120572

119894(120585 119906 119901) minus 119860

120572

119894(

120585 119901)

1003816

1003816

1003816

1003816

1003816

le 120581 (|119906|) 120583 (119889 (120585

120585) + |119906 minus |) (2)

for all 120585 120585 isin Ω 119906 isin R119873 and 119901 isin R2119899119873 where120581 (0 +infin) rarr [1 +infin) is monotone nondecreasing and120583 (0 +infin) rarr [0 +infin) is monotone nondecreasing andconcave with 120583(0+) = 0 We also required that 119903 rarr 119903

minus120574

120583(119903)

be nonincreasing for some 120574 isin (0 1) and that

119872(119903) = int

119903

0

120583 (120588)

120588

119889120588 lt infin for some 119903 gt 0 (3)

We adopt the method of A-harmonic approximation tothe case of sub-elliptic systems in the Heisenberg groupsand establish the optimal partial regularity result Roughlyspeaking assume additionally to the standard hypotheses(see precisely (H1) (H2) and (H4)) that (1+ |119901|)minus1119860120572

119894(120585 119906 119901)

satisfies (2) and (3) Let 119906 isin 11986711988212(ΩR119873) be aweak solutionof the sub-elliptic system (1) Then 119906 is of class 1198621 outsidea closed singular set Sing119906 sub Ω of the Haar measure 0Furthermore for 120585

0isin Ω Sing119906 the derivative 119883119906 of 119906 has

the modulus of continuity 119903 rarr 119872(119903) in a neighborhood of120585

0 Our result is optimal in the sense that in the case 120583(119903) = 119903120574

0 lt 120574 lt 1 we have119872(119903) = 120574minus1119903120574 Holder continuity Γ1120574 to beoptimal in that case

As is well known even under reasonable assumptionson 119860120572119894and 119861120572 of the systems of equations one cannot in

general expect that weak solutions of (1) will be classicalthat is 1198622-solutions This was first shown by de Giorgi [3]

2 Abstract and Applied Analysis

we also refer the reader to Giaquinta [4] and Chen and Wu[5] for further discussion and additional examples Thenthe goal is to establish partial regularity theory Moreover anew method calledA-harmonic approximation technique isintroduced by Duzaar and Steffen in [6] and simplified byDuzaar and Grotowski in [7] to study elliptic systems withquadratic growth caseThen similar results have been provedfor more general 119860120572

119894or 119861120572 in the Euclidean setting see [8ndash

11] for Holder continuous coefficients and [12ndash15] for Dinicontinuous coefficients

However turning to sub-elliptic equations and systemsin the Heisenberg groups H119899 some new difficulties willarise Already in the first step trying to apply the standarddifference quotient method the main difference betweenEuclidean R119899 and Heisenberg groups H119899 becomes clear Anytime we use horizontal difference quotients (ie in the direc-tions 119883

119894) extra terms with derivatives in the 119879 direction will

arise due to noncommutativity (see (12)) but these cannotbe controlled by using the initial assumptions on the weaksolution Several results were focused on those equationswhich have a bearing on basic vector fields on the Heisenberggroup or more generally the Carnot group Capogna [16 17]studied the regularities for weak solutions to quasi-linearequations Concretely by a technique combining fractionaldifference quotients and fractional derivatives defined byFourier transform differentiability in the nonhorizontaldirection11988222 estimate and119862infin continuity of weak solutionsare obtained see [16] for the case of Heisenberg groups and[17] for Carnot groups To sub-elliptic 119901-Laplace equations inHeisenberg groups Marchi in [18ndash20] showed that 119879119906 isin 119871119901locand 1198832119906 isin 119871

2

loc for 1 + (1radic5) lt 119901 lt 1 +

radic

5 by usingtheories of Besov space and Bessel potential space Domokosin [21 22] improved these results for 1 lt 119901 lt 4 employingthe A Zygmund theory related to vector fields Recently bymeticulous arguments Manfredi and Mingione in [23] andMingione et al in [24] proved Holder regularity with regardto full Euclidean gradient for weak solutions and further 119862infincontinuity under the coefficients assumed to be smooth

While regularities for weak solutions to sub-elliptic sys-tems concerning vector fields aremore complicated Capognaand Garofalo in [25] showed the partial Holder regularityfor the horizontal gradient of weak solutions to quasilinearsub-elliptic systems minussum119896

119894=1119883

119894(119860

120572

119894(120585 119906)119883

119895119906) = 119861

120572

(120585 119906 119883119906)

with 119883119894 119883119895(119894 119895 = 1 119896) being horizontal vector fields

in Carnot groups of step two where 119860120572119894and 119861120572 satisfy the

quadratic structure conditions Their way relies mainly ongeneralization of classical direct method in the Euclideansetting Shores in [26] considered a homogeneous quasi-linear system minussum

119896

119894=1119883

119894(119860

120572

119894(120585 119906)119883

119895119906) = 0 in the Carnot

group with general step where 119860120572119894also satisfies the quadratic

growth condition She established higher differentiability andsmoothness for weak solutions of the system with constantcoefficients and deduced partial regularity for weak solutionsof the original system With respect to the case of non-quadratic growth Foglein in [27] treated the homogeneousnonlinear system minussum

2119899

119894=1119883

119894119860

120572

119894(120585 119883119906) = 0 in the Heisenberg

group under superquadratic structure conditions She got

a priori estimates for weak solutions of the system withconstant coefficients and partial regularity for the horizontalgradient of weak solutions to the initial system Later Wangand Niu [1] and Wang and Liao [2] treated more generalnonlinear sub-elliptic system in the Carnot groups undersuperquadratic growth conditions and subquadratic growthconditions respectively

The regularity results for sub-elliptic systems mentionedabove require Holder continuity with respect to the coef-ficients 119860120572

119894 When the assumption of Holder continuity on

119860

120572

119894is weakened to Dini continuity how to establish partial

regularity of weak solutions to nonlinear sub-elliptic systemsin the Heisenberg group This paper is devoted to this topicTo define weak solution to (1) we assume the followingstructure conditions on 119860120572

119894and 119861120572

(H1) 119860120572119894(120585 119906 119901) is differentiable in 119901 and there exist some

constants 119871 such that1003816

1003816

1003816

1003816

1003816

1003816

1003816

119860

120572

119894119901119895

120573

(120585 119906 119901)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

le 119871 (120585 119906 119901) isin Ω timesR119873

timesR2119899119873

(4)

Here we write down 119860120572119894119901119895

120573

(120585 119906 119901) = (120597119860

120572

119894(120585 119906 119901)

120597119901

119895

120573)

(H2) 119860120572119894(120585 119906 119901) is uniformly elliptic that is for some 120582 gt

0 we have

119860

120572

119894119901119895

120573

(120585 119906 119901) 120578

120572

119894120578

120573

119895ge 120582

1003816

1003816

1003816

1003816

120578

1003816

1003816

1003816

1003816

2

forall120578 isin R2119899119873

(5)

(H3) There exist a modulus of continuity 120583 (0 +infin) rarr

[0 +infin) and a nondecreasing function 120581 [0 +infin) rarr

[1 +infin) such that

(1 +

1003816

1003816

1003816

1003816

119901

1003816

1003816

1003816

1003816

)

minus1 10038161003816

1003816

1003816

1003816

119860

120572

119894(120585 119906 119901) minus 119860

120572

119894(

120585 119901)

1003816

1003816

1003816

1003816

1003816

le 120581 (|119906|) 120583 (119889 (120585

120585) + |119906 minus |)

(6)

(H4) 119861120572 satisfies quadratic controllable growth condition

1003816

1003816

1003816

1003816

119861

120572

(120585 119906 119901)

1003816

1003816

1003816

1003816

le 119862 (1 + |119906|

119903minus1

+

1003816

1003816

1003816

1003816

119901

1003816

1003816

1003816

1003816

2(1minus1119903)

) (7)

where 119903 = 2119876(119876 minus 2) because 119876 gt 2 see (16)

Without loss of generality we can assume that 120581 ge 1 andthe following

(1205831) 120583 is nondecreasing with 120583(0+) = 0(1205832) 120583 is concave in the proof of the regularity theorem

we have to require that 119903 rarr 119903

minus120574

120583(119903) is nonincreasingfor some exponent 120574 isin (0 1) We also require Dinirsquoscondition (2) which was already mentioned in theintroduction

(1205833) 119872(119903) = int1199030

(120583(120588)120588)119889120588 lt infin for some 119903 gt 0

In the present paper we will apply the method of A-harmonic approximation adapting to the setting of Heisen-berg groups to study partial regularity for the system (1) Since

Abstract and Applied Analysis 3

basic vector fields 119883119894of Lie algebras corresponding to the

Heisenberg group are more complicated than gradient vectorfields in the Euclidean setting we have to find a differentauxiliary function in proving Caccioppoli type inequalityBesides the nonhorizontal derivatives of weak solutionswill happen in the Taylor type formula in the Heisenberggroup and cannot be effectively controlled in the presenthypotheses So the method employing Taylorrsquos formula in[12] is not appropriate in our setting In order to obtain thedesired decay estimate we use the Poincare type inequalityin [28] as a replacement And we obtain the following mainresult

Theorem 1 Assume that coefficients 119860120572119894and 119861120572 satisfy (H1)ndash

(H4) (1205831)ndash(1205833) and that 119906 isin 11986711988212(ΩR119873) is a weak solutionto the system (1) that is

int

Ω

119860

120572

119894(120585 119906 119883119906)119883

119894120601

120572

119889120585 = int

Ω

119861

120572

(120585 119906 119883119906) 120601

120572

119889120585

forall120601 isin 119862

infin

0(ΩR

119873

)

(8)

Then there exists a relatively closed set Sing 119906 sub Ω such that119906 isin 119862

1

(Ω Sing 119906 R119873) Furthermore Sing 119906 sub Σ1cup Σ

2and

Haar meas (Ω Sing 119906) = 0 where

Σ

1= 120585

0isin Ω sup

119903gt0

(

1003816

1003816

1003816

1003816

1003816

119906

1205850119903

1003816

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

1003816

(119883119906)

1205850119903

1003816

1003816

1003816

1003816

1003816

) = infin

Σ

2= 120585

0isin Ω lim

119903rarr0+inf 10038161003816

1003816

1003816

119861

119903(120585

0)

1003816

1003816

1003816

1003816

minus1

H119899

timesint

119861119903(1205850)

1003816

1003816

1003816

1003816

1003816

119883119906 minus (119883119906)

1205850119903

1003816

1003816

1003816

1003816

1003816

2

119889120585 gt 0

(9)

In addition for 120591 isin [120574 1) and 1205850isin Ω Sing 119906 the derivative

119883119906 has the modulus of continuity 119903 rarr 119903

120591

+ 119872(119903) in a neigh-borhood of 120585

0

2 Preliminaries

The Heisenberg group H119899 is defined as R2119899+1 endowed withthe following group multiplication

sdot H119899

times H119899

997888rarr H119899

((120585

1

119905) (

120585

1

119905)) 997891997888rarr (120585

1

+

120585

1

119905 +

119905 +

1

2

119899

sum

119894=1

(119909

119894119910

119894minus 119909

119894119910

119894))

(10)

for all 120585 = (120585

1

119905) = (119909

1 119909

2 119909

119899 119910

1 119910

2 119910

119899 119905) 120585 =

(

120585

1

119905) = (119909

1 119909

2 119909

119899 119910

1 119910

2 119910

119899

119905) This multiplicationcorresponds to addition in Euclidean R2119899+1 Its neutralelement is (0 0) and its inverse to (1205851 119905) is given by (minus1205851 minus119905)Particularly the mapping (120585 120585) 997891rarr 120585 sdot

120585

minus1 is smooth so (H119899 sdot)is a Lie group

The basic vector corresponding to its Lie algebra can beexplicitly calculated by the exponential map and is given by

119883

119894=

120597

120597119909

119894

minus

119910

119894

2

120597

120597119905

119883

119894+119899=

120597

120597119910

119894

+

119909

119894

2

120597

120597119905

119879 =

120597

120597119905

(11)

for 119894 = 1 2 119899 and note that the special structure of thecommutators

[119883

119894 119883

119894+119899] = minus [119883

119894+119899 119883

119894] = 119879 else [119883

119894 119883

119895] = 0

[119879 119879] = [119879119883

119894] = 0

(12)

that is (H119899 sdot) is a nilpotent Lie group of step 2 119883 =

(119883

1 119883

2119899) is called the horizontal gradient and 119879 the

vertical derivativeThe pseudonorm is defined by

1003817

1003817

1003817

1003817

1003817

(120585

1

119905)

1003817

1003817

1003817

1003817

1003817

= (

1003816

1003816

1003816

1003816

1003816

120585

11003816

1003816

1003816

1003816

1003816

4

+ 119905

2

)

14

(13)

and the metric induced by this pseudonorm is given by

119889 (

120585 120585) =

1003817

1003817

1003817

1003817

1003817

120585

minus1

sdot

120585

1003817

1003817

1003817

1003817

1003817

(14)

The measure used on H119899 is Haar measure and the volume ofthe pseudoball 119861

119877(120585

0) = 120585 isin H119899 119889(120585

0 120585) lt 119877 is given by

1003816

1003816

1003816

1003816

119861

119877(120585

0)

1003816

1003816

1003816

1003816H119899= 119877

2119899+210038161003816

1003816

1003816

119861

1(120585

0)

1003816

1003816

1003816

1003816H119899≜ 120596

119899119877

2119899+2

(15)

The number

119876 = 2119899 + 2 (16)

is called the homogeneous dimension of H119899The horizontal Sobolev spaces 1198671198821119901(Ω) (1 le 119901 lt infin)

are defined as

119867119882

1119901

(Ω) = 119906 isin 119871

119901

(Ω) 119883

119894119906 isin 119871

119901

(Ω)

119894 = 1 2 2119899

(17)

Then1198671198821119901(Ω) is a Banach space with the norm

119906

1198671198821119901(Ω)

= 119906

119871119901(Ω)+

2119899

sum

119894=1

1003817

1003817

1003817

1003817

119883

119894119906

1003817

1003817

1003817

1003817119871119901(Ω) (18)

119867119882

1119901

0(Ω) is the completion of 119862infin

0(Ω) under norm (18)

Lu [28] showed the following Poincare type inequalityrelated to Hormanderrsquos vector fields for 119906 isin 1198671198821119902(119861

119877(120585

0))

1 lt 119902 lt 119876 1 le 119901 le 119902119876(119876 minus 119902)

(∮

119861119877(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus 119906

1205850119877

1003816

1003816

1003816

1003816

1003816

119901

119889120585)

1119901

le 119862

119901119877(∮

119861119877(1205850)

|119883119906|

119902

119889120585)

1119902

(19)

where we write down ∮119861119903(1205850)

119906119889120585 = |119861

119903(120585

0)|

minus1

H119899 int119861119903(1205850)119906119889120585 here

and there Note the fact that the horizontal vectors119883119894defined

4 Abstract and Applied Analysis

in (11) fit Hormanderrsquos vector fields and that (19) is valid for119901 = 119902 = 2

Following [12] for technical convenience letting 120578(119905) =120583

2

(

radic

2119905) we have the corresponding properties for 120578 (1205781) 120578is continuous nondecreasing and 120578(0) = 0 (1205782) 120578 is concaveand 119903 rarr 119903

minus120574

120578(119903) is nonincreasing for some exponent 120574 isin(0 1) (1205783)119867(119903) = 41198722(radic2119903) = [int119903

0

(

radic

120578(120588)120588)119889120588]

2

lt infin forsome 119903 gt 0 Changing 120581 by a constant but keeping 120581 ge 1 wemay assume the following (1205784) 120578(1) = 1 implying 120578(119905) ge 119905for 119905 isin [0 1] Also note that it implies that from (1205782) and (1205784)120578(119905) le (120574

2

4)119867(119905) for all 119905 ge 0Furthermore the following inequality holds

119904120578 (119905) le 119904120578 (119904) + 119905 119904 isin [0 1] 119905 gt 0 (20)

The condition (H3) becomes

1003816

1003816

1003816

1003816

1003816

119860

120572

119894(120585 119906 119901) minus 119860

120572

119894(

120585 119901)

1003816

1003816

1003816

1003816

1003816

le 120581 (|119906|)radic120578 (119889

2

(120585

120585) + |119906 minus |

2

) (1 +

1003816

1003816

1003816

1003816

119901

1003816

1003816

1003816

1003816

)

(21)

Moreover we deduce the existence of a nonnegative modulusof continuity with 120596(119905 0) = 0 for all 119905 such that 120596(119904 119905) isnondecreasing with respect to 119905 for fixed 119904 and 1205962(119904 119905) isconcave and nondecreasing with respect to 119904 for fixed 119905 Alsowe have for |119906| + |119883119906| le 119872

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119860

120572

119894119901119895

120573

(120585 119906 119901) minus 119860

120572

119894119901119895

120573

(

120585

119901)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

le 120596 (119872 119889

2

(120585

120585) + |119906 minus |

2

+

1003816

1003816

1003816

1003816

119901 minus

119901

1003816

1003816

1003816

1003816

2

)

(22)

Using (H1) and (H2) we see that

1003816

1003816

1003816

1003816

119860

120572

119894(120585 119906 119901) minus 119860

120572

119894(120585 119906

119901)

1003816

1003816

1003816

1003816

le 119871

1003816

1003816

1003816

1003816

119901 minus

119901

1003816

1003816

1003816

1003816

(23)

(119860

120572

119894(120585 119906 119901) minus 119860

120572

119894(120585 119906

119901)) (119901 minus

119901) ge 120582

1003816

1003816

1003816

1003816

119901 minus

119901

1003816

1003816

1003816

1003816

2

(24)

In the sequel the constant 119862may vary from line to line

3 Caccioppoli Type Inequality

In this section we present the followingA-harmonic approx-imation lemma in the Heisenberg group introduced byFoglein [27] with 119901 = 2 as a special case and prove aCaccioppoli type inequality in our setting

Lemma 2 Let 120582 and 119871 be fixed positive numbers and 119899119873 isin

N with 119899 ge 2 If for any given 120576 gt 0 there exists 120575 =

120575(119899119873 120582 120576) isin (0 1] with the following properties

(I) for anyA isin Bil(R2119899119873) satisfying

A (] ]) ge 120582|]|2 A (] ]) le 119871 |]| |]| ] ] isin R2119899119873

(25)

(II) for any 119908 isin 11986711988212(119861120588(120585

0)R119873) satisfying

∮

119861120588(1205850)

|119883119908|

2

119889120585 le 1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

∮

119861120588(1205850)

A (119883119908119883120593) 119889120585

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

le 120575 sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

forall120593 isin 119862

1

0(119861

120588(120585

0) R119873

)

(26)

then there exists anA-harmonic function ℎ such that

∮

119861120588(1205850)

|119883ℎ|

2

119889120585 le 1 120588

minus2

∮

119861120588(1205850)

|ℎ minus 119908|

2

119889120585 le 120576 (27)

Foglein [27] established a priori estimate for the weaksolution 119906 to homogeneous sub-elliptic systemswith constantcoefficients in the Heisenberg group (also see [25] for Carnotgroups of step 2) We list it as follows

sup1198611205882(1205850)

(|119906|

2

+ 120588

2

|119883119906|

2

+ 120588

41003816

1003816

1003816

1003816

1003816

119883

2

119906

1003816

1003816

1003816

1003816

1003816

2

) le 119862

0∮

119861120588(1205850)

|119883119906|

2

119889120585

(28)

In what follows we let 1205881(119904 119905) = (1 + 119904 + 119905)

minus1

120581(119904 + 119905)

minus1 and119870

1(119904 119905) = (1 + 119905)

4

120581(119904 + 119905)

4 for 119904 119905 ge 0 Note that 1205881le 1 and

that 119904 rarr 120588

1(119904 119905) 119905 rarr 120588

1(119904 119905) are nonincreasing functions

Lemma 3 Let 119906 isin 11986711988212(ΩR119873) be a weak solution to thesystem (1) under the conditions (H1)ndash(H4) (1205831)ndash(1205833) Thenfor every 120585

0= (119909

0

1 119909

0

2 119909

0

119899 119910

0

1 119910

0

2 119910

0

119899 119905) isin Ω 119906

0isin R119873

119901

0isin R2119899119873 and 0 lt 120588 lt 119877 lt 120588

1(|119906

0| |119901

0|) le 1 such that

119861

119877(120585

0) subsub Ω the inequality

int

119861120588(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

119889120585

le 119862

119888[

1

(119877 minus 120588)

2int

119861119877(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus 119906

0minus (120585

1

minus 120585

1

0) 119901

0

1003816

1003816

1003816

1003816

1003816

2

119889120585 + 119865]

(29)

holds where 1205851 = (1199091 119909

2 119909

119899 119910

1 119910

2 119910

119899) is the horizon-

tal component of 120585 = (1205851 119905) isin Ω and

119865 = 120596

119899119877

119876

119870

1(

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

) 120578 (119877

2

)

+ [int

119861119877(1205850)

(1 + 119906

119903

+ |119883119906|

2

) 119889120585]

2(1minus1119903)

(30)

Proof Let V = 119906minus1199060minus(120585

1

minus120585

1

0)119901

0 Take a test function120593 = 1206012V

in (8) with 120601 isin 119862infin0(119861

119877(120585

0)R119873) satisfying 0 le 120601 le 1 |nabla120601| le

119862(119877minus120588) and 120601 equiv 1 on 119861120588(120585

0) Then we have119883V = 119883119906minus119901

0

|119883120593| le 120601|119883119906 minus 119901

0| + 119862(119877 minus 120588)|V| and

int

119861119877(1205850)

119860

120572

119894(120585 119906 119883119906) 120601

2

(119883119906 minus 119901

0) 119889120585

= minus2int

119861119877(1205850)

120601119883120601119860

120572

119894(120585 119906 119883119906) V119889120585

+ int

119861119877(1205850)

119861

120572

(120585 119906 119883119906) 120593

120572

119889120585

(31)

Abstract and Applied Analysis 5

Adding this to the equations

minus int

119861119877(1205850)

119860

120572

119894(120585 119906 119901

0) 120601

2

(119883119906 minus 119901

0) 119889120585

= 2int

119861119877(1205850)

120601119883120601119860

120572

119894(120585 119906 119901

0) V119889120585

minus int

119861119877(1205850)

119860

120572

119894(120585 119906 119901

0)119883120593

120572

119889120585

0 = int

119861119877(1205850)

119860

120572

119894(120585

0 119906

0 119901

0)119883120593

120572

(32)

It follows that by using the hypotheses (H1) (H3) (ie (23)(21) resp) and (H4)

int

119861119877(1205850)

[119860

120572

119894(120585 119906 119883119906)

minus119860

120572

119894(120585 119906 119901

0)] 120601

2

(119883119906 minus 119901

0) 119889120585

= 2int

119861119877(1205850)

[119860

120572

119894(120585 119906 119901

0)

minus119860

120572

119894(120585 119906 119883119906)] 120601V119883120601119889120585

+ int

119861119877(1205850)

[119860

120572

119894(120585 119906

0+ (120585

1

minus 120585

1

0) 119901

0 119901

0)

minus119860

120572

119894(120585 119906 119901

0) ]119883120593

120572

119889120585

+ int

119861119877(1205850)

[119860

120572

119894(120585

0 119906

0 119901

0)

minus119860

120572

119894(120585 119906

0+ (120585

1

minus 120585

1

0) 119901

0 119901

0)]119883120593

120572

119889120585

+ int

119861119877(1205850)

119861

120572

(120585 119906 119883119906) 120593

120572

119889120585

le 119868 + 119868119868 + 119868119868119868 + 119868119881 + 119881

(33)

where

119868 = 2119871int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

120601

1003816

1003816

1003816

1003816

|V| 1003816100381610038161003816

119883120601

1003816

1003816

1003816

1003816

119889120585

119868119868 = (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

) 120581 (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+ 119877

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

times int

119861119877(1205850)

radic120578 (|V|2) 100381610038161003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

120601

2

119889120585

119868119868119868 = 2 (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

) 120581 (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+ 119877

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

times int

119861119877(1205850)

radic120578 (|V|2) |V| 100381610038161003816

1003816

119883120601

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

120601

1003816

1003816

1003816

1003816

119889120585

119868119881 = (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

) 120581 (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+ 119877

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

times int

119861119877(1205850)

radic120578 (119877

2

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

2

)) [

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

120601

2

+2

1003816

1003816

1003816

1003816

120601

1003816

1003816

1003816

1003816

|V| 1003816100381610038161003816

119883120601

1003816

1003816

1003816

1003816

] 119889120585

119881 = 119862int

119861119877(1205850)

(1 + |119906|

119903minus1

+ |119883119906|

2(1minus1119903)

) 120593119889120585

(34)

Applying (H2) the left hand side of (33) can be estimated as

120582int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

120601

2

119889120585

le int

119861119877(1205850)

[119860

120572

119894(120585 119906 119883119906) minus 119860

120572

119894(120585 119906 119901

0)] 120601

2

(119883119906 minus 119901

0) 119889120585

(35)

For 120576 gt 0 to be fixed later we have using Youngrsquos inequality

119868 le 120576int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

21003816

1003816

1003816

1003816

120601

1003816

1003816

1003816

1003816

2

119889120585 +

119862119871

2

(119877 minus 120588)

2int

119861119877(1205850)

|V|2119889120585

(36)

Using Jensenrsquos inequality (20) and the fact that 120578(119905119904) le 119905120578(119904)for 119905 ge 1 we arrive at

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

120581

2

(sdot) int

119861119877(1205850)

120578 (|V|2) 119889120585

le 120596

119899119877

119876minus2

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

120581

2

(sdot) 119877

2

120578 (∮

119861119877(1205850)

|V|2119889120585)

le 120596

119899119877

119876minus2

[∮

119861119877(1205850)

|V|2119889120585

+ (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

120581

2

(sdot) 119877

2

120578

times ((1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

120581

2

(sdot) 119877

2

) ]

le 119877

minus2

int

119861119877(1205850)

|V|2119889120585

+ 120596

119899119877

119876

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

4

120581

4

(sdot) 120578 (119877

2

)

(37)

where 120581(sdot) is an abbreviation of the function 120581(|1199060| + |119901

0|)

Also note that the application of (20) in the second lastinequality is possible by our choice 119877 le 120588

1(|119906

0| + |119901

0|)

6 Abstract and Applied Analysis

Using Youngrsquos inequality and (37) in 119868119868 we obtain

119868119868 le 120576int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

21003816

1003816

1003816

1003816

120601

1003816

1003816

1003816

1003816

2

119889120585

+ 120576

minus1

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

120581

2

(sdot) int

119861119877(1205850)

120578 (|V|2) 119889120585

le 120576int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

21003816

1003816

1003816

1003816

120601

1003816

1003816

1003816

1003816

2

119889120585

+

1

120576(119877 minus 120588)

2int

119861119877(1205850)

|V|2119889120585

+ 120576

minus1

120596

119899119877

119876

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

4

120581

4

(sdot) 120578 (119877

2

)

(38)

And similarly we see

119868119868119868 le

4119862

(119877 minus 120588)

2int

119861119877(1205850)

|V|2119889120585

+ (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

120581

2

(sdot) int

119861119877(1205850)

120578 (|V|2) 119889120585

le

119862

(119877 minus 120588)

2int

119861119877(1205850)

|V|2119889120585

+ 120596

119899119877

119876

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

4

120581

4

(sdot) 120578 (119877

2

)

119868119881 le 120576int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

21003816

1003816

1003816

1003816

120601

1003816

1003816

1003816

1003816

2

119889120585

+

4119862120576

(119877 minus 120588)

2int

119861119877(1205850)

|V|2119889120585

+ 120576

minus1

120596

119899119877

119876

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

120581

2

(sdot) 120578

times (∮

119861119877(1205850)

119877

2

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

2

) 119889120585)

le 120576int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

21003816

1003816

1003816

1003816

120601

1003816

1003816

1003816

1003816

2

119889120585

+

119862120576

(119877 minus 120588)

2int

119861119877(1205850)

|V|2119889120585

+ 120576

minus1

120596

119899119877

119876

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

4

120581

4

(sdot) 120578 (119877

2

)

(39)

Here we have used 120581 ge 1 in the last inequality

By Holderrsquos inequality (19) and Youngrsquos inequality onegets

119881 le 119862(int

119861119877(1205850)

1003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119903

119889120585)

1119903

times (int

119861119877(1205850)

(1 + |119906|

119903

+ |119883119906|

2

) 119889120585)

(119903minus1)119903

le 119862(int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

2

119889120585)

12

times (int

119861119877(1205850)

(1 + |119906|

119903

+ |119883119906|

2

) 119889120585)

(119903minus1)119903

le 120576int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

2

119889120585 + 119862 (120576)

times (int

119861119877(1205850)

(1 + |119906|

119903

+ |119883119906|

2

) 119889120585)

2(119903minus1)119903

le 120576int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

21003816

1003816

1003816

1003816

120601

1003816

1003816

1003816

1003816

2

119889120585

+

119862120576

(119877 minus 120588)

2int

119861119877(1205850)

|V|2119889120585

+ 119862 (120576) (int

119861119877(1205850)

(1 + |119906|

119903

+ |119883119906|

2

) 119889120585)

2(119903minus1)119903

(40)

where we have used the fact that |119883120593| le 120601|119883119906 minus 1199010| + 119862(119877 minus

120588)|V|Applying these estimates to (37) we obtain

(120582 minus 4120576) int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

120601

2

119889120585

le

119862 (119871 120576)

(119877 minus 120588)

2int

119861119877(1205850)

|V|2119889120585

+ (120576

minus1

+ 2)120596

119899119877

119876

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

4

120581

4

(sdot) 120578 (119877

2

)

+ 119862 (120576) (int

119861119877(1205850)

(1 + |119906|

119903

+ |119883119906|

2

) 119889120585)

2(119903minus1)119903

(41)

Choosing 120576 = 1205828 we obtain the desired inequality (29)

4 Proof of the Main Theorem

In this section we will complete the proof of the partialregularity results via the following lemmas In the sequel wealways suppose that 119906 isin 11986711988212(ΩR119873) is a weak solution to(1) with the assumptions of (H1)ndash(H4) and (1205831)ndash(1205833)

Abstract and Applied Analysis 7

Lemma 4 Let 119861120588(120585

0) subsub Ω with 120588 le 120588

1(|119906

0| |119901

0|) and 120593 isin

119862

infin

0(119861

120588(120585

0)R119873) satisfying |120593| le 120588

2 and sup119861120588(1205850)

|119883120593| le 1Then there exists a constant 119862

1ge 1 such that

∮

119861120588(1205850)

119860

120572

119894119901119895

120573

(120585

0 119906

0 119901

0) (119883119906 minus 119901

0)119883120593

120572

119889120585

le 119862

1[Φ (120585

0 120588 119901

0)

+ 120596 (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

Φ (120585

0 120588 119901

0))Φ

12

(120585

0 120588 119901

0)

+ 119870

1(

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)radic120578 (120588

2

)] sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

(42)

Proof Using the fact that int119861120588(1205850)

119860

120572

119894(120585

0 119906

0 119901

0)119883120593

120572

119889120585 = 0 andthe weak form (8) we deduce

∮

119861120588(1205850)

[int

1

0

119860

120572

119894119901119895

120573

(120585

0 119906

0 120579119883119906 + (1 minus 120579) 119901

0)

times (119883119906 minus 119901

0) 119889120579]119883120593

120572

119889120585

= ∮

119861120588(1205850)

[119860

120572

119894(120585

0 119906

0 119883119906)

minus119860

120572

119894(120585

0 119906

0 119901

0)]119883120593

120572

119889120585

= ∮

119861120588(1205850)

[119860

120572

119894(120585

0 119906

0 119883119906)

minus119860

120572

119894(120585 119906 119883119906)]119883120593

120572

119889120585

+ ∮

119861120588(1205850)

119861

120572

(120585 119906 119883119906) 120593

120572

119889120585

(43)

It yields

∮

119861120588(1205850)

119860

120572

119894119901119895

120573

(120585

0 119906

0 119901

0) (119883119906 minus 119901

0)119883120593

120572

119889120585

= ∮

119861120588(1205850)

[int

1

0

(119860

120572

119894119901119895

120573

(120585

0 119906

0 119901

0)

minus119860

120572

119894119901119895

120573

(120585

0 119906

0 120579119883119906 + (1 minus 120579) 119901

0))

times (119883119906 minus 119901

0) 119889120579] 119889120585 sup

119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

+ ∮

119861120588(1205850)

[119860

120572

119894(120585

0 119906

0 119883119906)

minus119860

120572

119894(120585 119906

0+ 119901

0(120585 minus 120585

0) 119883119906)] sup

119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

+ ∮

119861120588(1205850)

[119860

120572

119894(120585 119906

0+ 119901

0(120585 minus 120585

0) 119883119906)

minus119860

120572

119894(120585 119906 119883119906)] sup

119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

+ ∮

119861120588(1205850)

119861

120572

(120585 119906 119883119906) 120593

120572

119889120585

= 119868

1015840

+ 119868119868

1015840

+ 119868119868119868

1015840

+ 119868119881

1015840

(44)

Using (22) Holderrsquos inequality the fact that 119905 rarr 120596

2

(119904 119905) isconcave and Jensenrsquos inequality we have

119868

1015840

le sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

∮

119861120588(1205850)

120596 (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

119889120585

le sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

[∮

119861120588(1205850)

120596

2

(

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

) 119889120585]

12

times[∮

119861120588(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

119889120585]

12

le 120596(

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

Φ (120585

0 120588 119901

0))Φ

12

(120585

0 120588 119901

0) sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

(45)

Similarly using (21) and the fact that 120578(119905119904) le 119905120578(119904) for 119905 ge 1we obtain

119868119868

1015840

le sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

120581 (sdot)radic120578 (120588

2

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

2

))

times ∮

119861120588(1205850)

(1 + |119883119906|) 119889120585

le sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

120581 (sdot)radic120578 (120588

2

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

)

times ∮

119861120588(1205850)

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

) 119889120585

le sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

[(∮

119861120588(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

119889120585)

+ 120581

2

(sdot) (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

120578 (120588

2

)

+ 120581 (sdot) (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

3

radic120578 (120588

2

) ]

le [Φ (120585

0 120588 119901

0)+ 2120581

2

(sdot) (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

3

radic120578 (120588

2

)]

times sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

(46)

8 Abstract and Applied Analysis

where we have used the fact that 120578(1205882) le

radic

120578(120588

2

) whichfollows from the nondecreasing property of the function 120578(119905)(1205784) and our assumption 120588 le 120588

1le 1

In the same way it follows that by using (21) (37) and(19)

119868119868119868

1015840

le sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

120581 (sdot)∮

119861120588(1205850)

radic120578 (|V|2) (1 + |119883119906|) 119889120585

le sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

[∮

119861120588(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

119889120585

+ 120581

2

(sdot)∮

119861120588(1205850)

120578 (|V|2) 119889120585

+ 120581 (sdot) (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)∮

119861120588(1205850)

radic120578 (|V|2) 119889120585]

le sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

[Φ (120585

0 120588 119901

0)

+ 2120588

minus2

∮

119861120588(1205850)

|V|2119889120585 + 1205814 (sdot) 120578 (1205882)

+ 120581

2

(sdot) (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

radic120578 (120588

2

) ]

le [(1 + 2119862

119901)Φ (120585

0 120588 119901

0)

+ 2120581

4

(sdot) (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

radic120578 (120588

2

)]

(47)

Using Holderrsquos inequality (19) and Youngrsquos inequality wehave

119868119881

1015840

le 119862∮

119861120588(1205850)

(1 + |119906|

119903minus1

+ |119883119906|

2(1minus1119903)

)

1003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119889120585

le 119862∮

119861120588(1205850)

|119883119906|

2(1minus1119903) 10038161003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119889120585

+ 119862∮

119861120588(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus 119906

0minus 119901

0(120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

119903minus11003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119889120585

+ 119862120588

2

[1 + (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

119903minus1

]

le 119862(∮

119861120588(1205850)

|119883119906|

2

119889120585)

(1minus1119903)

(∮

119861120588(1205850)

1003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119903

119889120585)

1119903

+ 119862(∮

119861120588(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus 119906

0minus 119901

0(120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

119903

119889120585)

(1minus1119903)

times (∮

119861120588(1205850)

1003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119903

119889120585)

1119903

+ 119862120588

2

[1 + (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

119903minus1

]

le 119862(∮

119861120588(1205850)

|119883119906|

2

119889120585)

(1minus1119903)

(∮

119861120588(1205850)

1003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119903

119889120585)

1119903

+ (∮

119861120588(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

119889120585)

(1199032)(1minus1119903)

times (∮

119861120588(1205850)

1003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119903

119889120585)

1119903

+ 119862120588

2

[1 + (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

119903minus1

]

le 119862119906

11986711988212(∮

119861120588(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

119889120585)

(1minus1119903)

times (∮

119861120588(1205850)

1003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119903

119889120585)

1119903

+ 119862120588 [1 + (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

119903minus1

]

le 119862119906

11986711988212(119861120588(1205850))

(∮

119861120588(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

119889120585)

+ 119862120588

2119903

+ 119862120588

2

[1 + (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

119903minus1

]

le 119862

2Φ(120585

0 120588 119901

0) + 119862120588

2

(1 +

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

119903minus1

le 119862

2Φ(120585

0 120588 119901

0) + 119862120581 (sdot) (1 +

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

radic120578 (120588

2

)

(48)

where we have used the assumption (1205784) and the fact that 119903 =2119876(119876 minus 2) = (2119899 + 4)2119899 le 3 and 119862

2= 119862119906

11986711988212(119861120588(1205850))

ge

1 Combining these estimates we obtain the conclusion with119862

1= (1 + 119862

2+ 2119862

119901) ge 1

Lemma 5 Assume that the conditions of Lemma 2 and thefollowing smallness conditions hold

120596 (

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

1003816

(119883119906)

1205850120588

1003816

1003816

1003816

1003816

1003816

Φ (120585

0 120588 (119883119906)

1205850120588))

+ Φ

12

(120585

0 120588 (119883119906)

1205850120588) le

120575

2

(49)

119862

3119870

2

1(

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

1205850120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

) le 120575

2 (50)

with 1198623= 8119862

2

1119862

5 together with

120588 le 120588

1(1 +

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

1 +

1003816

1003816

1003816

1003816

1003816

(119883119906)

1205850120588

1003816

1003816

1003816

1003816

1003816

) (51)

Then the following growth condition holds for 120591 isin [120574 1)

Φ (120585

0 120579120588) le 120579

2120591

Φ(120585

0 120588)

+ 119870

lowast

(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

)

(52)

where one abbreviates Φ(1205850 119903) = Φ(120585

0 119903 (119883119906)

1205850119903) and

119870

lowast

(119904 119905) = 119870(119904 119905) + (2 + 119904 + 119905)

2(119903minus1) with 119870(119904 119905) = (4120575

minus2

+

2

119876

119862

119888)119870

2

1(1 + 119904 1 + 119905)

Proof We define 119908 = [119906 minus 1199061205850120588

minus 119901

0(120585

1

minus 120585

1

0)]120590

minus1 where

120590 = 119862

1radic

Φ(120585

0 120588 119901

0) + 4120575

minus2119870

2

1(

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

) 120578 (120588

2)

(53)

Abstract and Applied Analysis 9

Then we have 119883119908 = 120590

minus1

(119883119906 minus 119901

0) Now we consider

119861

120588(120585

0) subsub Ω such that 120588 le 120588

1(|119906

0| |119901

0|) Applying Lemma 4

on 119861120588(120585

0) to 119906 we have for any 120593 isin 119862infin

0(119861

120588(120585

0)R119873)

∮

119861120588(1205850)

|119883119908|

2

119889120585 = 120590

minus2

Φ(120585

0 120588 119901

0) le

1

119862

2

1

le 1 (54)

∮

119861120588(1205850)

119860

120572

119894119901119895

120573

(120585

0 119906

0 119901

0)119883119908119883120593119889120585

le [Φ

12

(120585

0 120588 119901

0)

+120596 (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

Φ (120585

0 120588 119901

0)) +

120575

2

] sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

(55)

In consideration of the small condition (49) we see that(54) and (55) imply conditions (26) in Lemma 2 Also notethat (H1) and (H3) imply condition (25) So there exists an119860

120572

119894119901119895

120573

(120585

0 119906

0 119901

0)-harmonic function ℎ isin 11986711988212(119861

120588(120585

0)R119873)

such that

∮

119861120588(1205850)

|119883ℎ|

2

119889120585 le 1 120588

minus2

∮

119861120588(1205850)

|119908 minus ℎ|

2

119889120585 le 120576 (56)

Taking 1199060= 119906

12058502120579120588 120579 isin (0 14] and replacing 119901

0by 1199010+

120590(119883ℎ)

1205850 2120579120588 we use Lemma 3 to obtain

int

119861120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119883119906 minus 119901

0minus 120590(119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

2

119889120585

le 119862

119888[

1

(120579120588)

2int

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus 119906

12058502120579120588

minus (119901

0+ 120590(119883ℎ)

12058502120579120588)

times (120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

2

119889120585 + 119865]

(57)

where

119865 = 120596

119899(2120579120588)

119876

119870

1(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119901

0+ 120590(119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 ((2120579120588)

2

)

+ [int

1198612120579120588(1205850)

(1 + 119906

119903

+ |119883119906|

2

) 119889120585]

2(1minus1119903)

(58)

Using the fact that 119906 minus (1199010+ 120590(119883ℎ)

12058502120579120588)(120585

1

minus 120585

1

0) has mean

value 11990612058502120579120588

on the ball 1198612120579120588(120585

0) the definition of119908 and (19)

we have

1

(120579120588)

2∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus 119906

1205850 2120579120588minus (119901

0+ 120590(119883ℎ)

12058502120579120588) (120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

2

119889120585

le

4120590

2

(2120579120588)

2∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119908 minus ℎ

1205850 2120579120588

minus(119883ℎ)

12058502120579120588(120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

2

119889120585

le

4120590

2

(2120579120588)

2[∮

1198612120579120588(1205850)

|119908 minus ℎ|

2

119889120585

+ ∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

ℎ minus ℎ

1205850 2120579120588

minus(119883ℎ)

12058502120579120588(120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

2

119889120585]

le 4120590

2

[(2120579)

minus119876minus2

120576 + 119862

119901∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119883ℎ minus (119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

2

119889120585]

le 4120590

2

[(2120579)

minus119876minus2

120576 + 119862

2

119901(2120579120588)

2

∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119883

2

ℎ

1003816

1003816

1003816

1003816

1003816

2

119889120585]

le 4120590

2

[(2120579)

minus119876minus2

120576 + 119862

2

119901(2120579)

2

119862

0]

le 119862

4(120579

minus119876minus2

120576 + 120579

2

) [Φ (120585

0 120588 119901

0)

+ 4120575

minus2

119870

2

1(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

) 120578 (120588

2

)]

(59)

where 1198624= 119862

4(119876 120582 119871) ge 1 Note that in the second last

inequality we have used the fact that

∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119883

2

ℎ

1003816

1003816

1003816

1003816

1003816

119889120585 le sup119861120588(1205850)

1003816

1003816

1003816

1003816

1003816

119883

2

ℎ

1003816

1003816

1003816

1003816

1003816

le 119862

0120588

minus2

∮

119861120588(1205850)

|119883ℎ|

2

119889120585 le 119862

0120588

minus2

(60)

In consideration of the fact that 119903 = 2119876(119876 minus 2) gt 2 119876 ge 4

and the assumptions 120579 isin (0 14] and Φ le 1 it follows that

[∮

1198612120579120588(1205850)

(1 + 119906

119903

+ |119883119906|

2

) 119889120585]

2(1minus1119903)

le 119862[∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

119889120585]

2(1minus1119903)

+ 119862(∮

1198612120579120588(1205850)

|119883119906|

2

119889120585)

119903minus1

+ (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

4(1minus1119903)

)

10 Abstract and Applied Analysis

le 119862 [(2120579)

minus2119876(1minus1119903)

Φ(120585

0 120588 119901

0)

2(1minus1119903)

+(2120579)

minus119876(119903minus1)

Φ(120585

0 120588 119901

0)

119903minus1

]

+ (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

2(1minus1119903)

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

4(1minus1119903)

)

le 119862(2120579)

minus119876(119903minus1)

Φ(120585

0 120588 119901

0)

2(1minus1119903)

+ (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

2(1minus1119903)

)

2

(61)

Let 119875 = 1199010+120590(119883ℎ)

12058502120579120588with 119901

0= (119883119906)

12058502120579120588 Combining

these estimates (57)ndash(61) and considering the small condition(51) (it implies 120588 le 120588

1(|119906

12058502120579120588| |119875|) see (64) and (65)) we

deduce that

Φ(120585

0 120579120588) le

1003816

1003816

1003816

1003816

1003816

119861

120579120588(120585

0)

1003816

1003816

1003816

1003816

1003816

minus1

H119899int

1198612120579120588(1205850)

|119883119906 minus 119875|

2

119889120585

le 119862

119888

2

119876

(120579120588)

2∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus 119906

12058502120579120588

minus (119901

0+ 120590(119883ℎ)

12058502120579120588)

times (120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

2

119889120585

+ 2

119876

119862

119888119870

1(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119901

0+ 120590(119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

)

+ 119862

119888

(2120579120588)

2119876(1minus1119903)

(120579120588)

119876

times[∮

1198612120579120588(1205850)

(1 + 119906

119903

+ |119883119906|

2

) 119889120585]

2(1minus1119903)

le 2

119876

119862

4119862

119888(120579

minus119876minus2

120576 + 120579

2

)

times [Φ (120585

0 120588) + 4120575

minus2

119870

2

1

times (

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

) ]

+ 2

119876

119862

119888119870

1(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

+120590(119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

)

+ [2

119876

119862

119888(2120579)

2minus119876(119903minus1)

Φ(120585

0 120588)

2(1minus1119903)

+(1 +

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

2(1minus1119903)

)

2

] 120588

2

(62)

Wenow specify 120576 = 120579119876+4 120579 isin (0 14] such that 2119876+11198624119862

119888120579

2

le

120579

2120591 Note that the small condition (50) implies 12059021198625le 1 with

119862

5= max119862

0 119862

1198882

119876

(2120579)

minus(1198762+4)(119876minus2)

and it yields

2

119876

119862

119888(2120579)

2minus119876(119903minus1)

Φ(120585

0 120588)

2(1minus1119903)

le 1

(63)1003816

1003816

1003816

1003816

1003816

120590(119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

le 120590 sup1198612120579120588(1205850)

|119883ℎ|

le 120590radic119862

0(∮

119861120588(1205850)

|119883ℎ|

2

119889120585) le 120590radic119862

0le 1

(64)

where we have used the a priori estimate (28) for the A-harmonic function ℎ Furthermore using (19) and recallingthe definition of 120590 and 119862

1 we have

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

le

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588minus 119906

1205850120588

1003816

1003816

1003816

1003816

1003816

le

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+ (2120579)

minus1198762

times (∮

119861120588(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus (119883119906)

1205850120588(120585

1

minus 120585

1

0) minus 119906

1205850120588

1003816

1003816

1003816

1003816

1003816

2

119889120585)

12

le

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+ (2120579)

minus1198762

120588radic119862

119901Φ

12

(120585

0 120588)

le

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+

120590

radic

119862

119901

119862

1(2120579)

1198762

le

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+ 120590radic119862

5le

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+ 1

(65)

Combining these estimates with (62) we have

Φ(120585

0 120579120588) le 120579

2120591

Φ(120585

0 120588)

+ [4120575

minus2

119870

2

1(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

)

+ 2

119876

119862

119888119870

1(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

+120590(119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

)] 120578 (120588

2

)

+ [1 + (1 +

1003816

1003816

1003816

1003816

1003816

(119883119906)

1205850 2120579120588

1003816

1003816

1003816

1003816

1003816

2(1minus1119903)

)

2

] 120578 (120588

2

)

le 120579

2120591

Φ(120585

0 120588) + 119870 (

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

)

+ (2 +

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

)

2(119903minus1)

120578 (120588

2

)

le 120579

2120591

Φ(120585

0 120588) + 119870

lowast

(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

)

(66)

Then the proof of Lemma 5 is complete

Abstract and Applied Analysis 11

For 119879 gt 0 we find Φ0(119879) gt 0 (depending on 119876119873 120582 119871

120591 and 120596) such that

120596

2

(2119879 2Φ

0(119879)) + 2Φ

0(119879) le

1

2

120575

2

119862

1Φ

0(119879) le 120579

119876

(1 minus 120579

120591

)

2

119879

2

(67)

With Φ0(119879) from (67) we choose 120588

0(119879) isin (0 1] (depending

on 119876119873 120582 119871 120591 120596 120578 and 120581) such that

120588

0(119879) le 120588

1(1 + 2119879 1 + 2119879)

119862

3119870

2

1(2119879 2119879) 120578 (120588

0(119879)

2

) le 120575

2

119870

0(119879) 120578 (120588

0(119879)

2

) le (120579

2120574

minus 120579

2120591

)Φ

0(119879)

2 (1 + 119862

119901)119870

0(119879)119867 (120588

0(119879)

2

) le 120579

119876

(1 minus 120579

120574

)

2

(120579

2120574

minus 120579

2120591

) 119879

2

(68)

where1198700(119879) = 119870

lowast

(2119879 2119879)By the proof method of of Lemma 51 in [12] and

conditions (67)-(68) Lemma 6 can be proved As is wellknown it is sufficient to complete the proof ofTheorem 1 oncewe obtain Lemma 6

Lemma 6 Assume that for some 1198790gt 0 and 119861

120588(120585

0) subsub Ω one

has

(1) |1199061205850120588| + |(119883119906)

1205850120588| le 119879

0

(2) 120588 le 1205880(119879

0)

(3) Φ(1205850 120588) le Φ

0(119879

0)

Then the small conditions (49)ndash(51) are satisfied on the balls119861

120579119895120588(120585

0) for 119895 isin 119873 cup 0 Moreover the limit Λ

1205850=

lim119895rarrinfin

(119883119906)

1205850120579119895120588exists and the inequality

∮

119861120588(1205850)

1003816

1003816

1003816

1003816

1003816

119883119906 minus Λ

1205850

1003816

1003816

1003816

1003816

1003816

2

119889120585 le 119862

6((

119903

120588

)

2120591

Φ(120585

0 120588) + 119867 (119903

2

))

(69)

is valid for 0 lt 119903 le 120588 with a constant 1198626= 119862

6(119876 119873 120582 119871

120591 119886119899119889 119879

0)

Proof The proof is very similar to the proof of Lemma 51 in[12] We omit it here

Acknowledgments

The project was supported by the National Natural ScienceFoundation of China (no 11201081 and no 11126294) andby the Science and Technology Planning Project of JiangxiProvince China no GJJ13657

References

[1] J Wang and P Niu ldquoOptimal partial regularity for weaksolutions of nonlinear sub-elliptic systems in Carnot groupsrdquoNonlinear Analysis Theory Methods amp Applications vol 72 no11 pp 4162ndash4187 2010

[2] J Wang and D Liao ldquoOptimal partial regularity for sub-elliptic systems with sub-quadratic growth in Carnot groupsrdquoNonlinear Analysis Theory Methods amp Applications vol 75 no4 pp 2499ndash2519 2012

[3] E de Giorgi ldquoUn esempio di estremali discontinue per unproblema variazionale di tipo ellitticordquo Bollettino della UnioneMatematica Italiana vol 4 pp 135ndash137 1968

[4] M Giaquinta Multiple Integrals in the Calculus of Variationsand Nonlinear Elliptic Systems Princeton University PressPrinceton NJ USA 1983

[5] Y Chen and L Wu Second Order Elliptic Equations and EllipticSystems Science Press Beijing China 2003

[6] F Duzaar and K Steffen ldquoOptimal interior and boundary reg-ularity for almost minimizers to elliptic variational integralsrdquoJournal fur die Reine und Angewandte Mathematik vol 546 pp73ndash138 2002

[7] F Duzaar and J F Grotowski ldquoOptimal interior partial regu-larity for nonlinear elliptic systems the method of A-harmonicapproximationrdquo Manuscripta Mathematica vol 103 no 3 pp267ndash298 2000

[8] F Duzaar J F Grotowski and M Kronz ldquoRegularity ofalmost minimizers of quasi-convex variational integrals withsubquadratic growthrdquo Annali di Matematica Pura ed ApplicataIV vol 184 no 4 pp 421ndash448 2005

[9] F Duzaar and G Mingione ldquoThe p-harmonic approximationand the regularity of p-harmonic mapsrdquo Calculus of Variationsand Partial Differential Equations vol 20 no 3 pp 235ndash2562004

[10] F Duzaar and G Mingione ldquoRegularity for degenerate ellipticproblems via p-harmonic approximationrdquo Annales de lrsquoInstitutHenri Poincare Analyse Non Lineaire vol 21 no 5 pp 735ndash7662004

[11] S Chen andZ Tan ldquoThemethod ofA-harmonic approximationand optimal interior partial regularity for nonlinear ellipticsystems under the controllable growth conditionrdquo Journal ofMathematical Analysis and Applications vol 335 no 1 pp 20ndash42 2007

[12] F Duzaar and A Gastel ldquoNonlinear elliptic systems with Dinicontinuous coefficientsrdquo Archiv der Mathematik vol 78 no 1pp 58ndash73 2002

[13] F Duzaar A Gastel and G Mingione ldquoElliptic systemssingular sets and Dini continuityrdquo Communications in PartialDifferential Equations vol 29 no 7-8 pp 1215ndash1240 2004

[14] Y Qiu and Z Tan ldquoOptimal interior partial regularity fornonlinear elliptic systems with Dini continuous coefficientsrdquoActa Mathematica Scientia B vol 30 no 5 pp 1541ndash1554 2010

[15] Y Qiu ldquoOptimal partial regularity of second order nonlin-ear elliptic systems with Dini continuous coefficients for thesuperquadratic caserdquo Nonlinear Analysis Theory Methods ampApplications vol 75 no 8 pp 3574ndash3590 2012

[16] L Capogna ldquoRegularity of quasi-linear equations in theHeisen-berg grouprdquoCommunications on Pure andAppliedMathematicsvol 50 no 9 pp 867ndash889 1997

[17] L Capogna ldquoRegularity for quasilinear equations and 1-quasiconformal maps in Carnot groupsrdquo MathematischeAnnalen vol 313 no 2 pp 263ndash295 1999

[18] S Marchi ldquo1198621120572 local regularity for the solutions of the p-Laplacian on the Heisenberg group for 2 lt 119901 lt 1 +

radic

5rdquoZeitschrift fur Analysis und ihre Anwendungen vol 20 no 3 pp617ndash636 2001

12 Abstract and Applied Analysis

[19] S Marchi ldquo1198621120572 local regularity for the solutions of the p-Laplacian on the Heisenberg groupThe case 1+1radic5 lt 119901 le 2rdquoCommentationes Mathematicae Universitatis Carolinae vol 44no 1 pp 33ndash56 2003

[20] S Marchi ldquoL119901 regularity of the derivative in the secondcommutator direction for nonlinear elliptic equations on theHeisenberg grouprdquo Accademia Nazionale delle Scienze detta deiXL Rendiconti Serie V Memorie di Matematica e ApplicazioniParte I vol 26 pp 1ndash15 2002

[21] A Domokos ldquoDifferentiability of solutions for the non-degenerate p-Laplacian in the Heisenberg grouprdquo Journal ofDifferential Equations vol 204 no 2 pp 439ndash470 2004

[22] A Domokos On the regularity of p-harmonic functions inthe Heisenberg group [PhD thesis] University of PittsburghPittsburgh Pa USA 2004

[23] J J Manfredi and G Mingione ldquoRegularity results for quasilin-ear elliptic equations in the Heisenberg grouprdquo MathematischeAnnalen vol 339 no 3 pp 485ndash544 2007

[24] G Mingione A Zatorska-Goldstein and X Zhong ldquoGradientregularity for elliptic equations in the Heisenberg grouprdquoAdvances in Mathematics vol 222 no 1 pp 62ndash129 2009

[25] L Capogna and N Garofalo ldquoRegularity of minimizers of thecalculus of variations inCarnot groups via hypoellipticity of sys-tems ofHormander typerdquo Journal of the EuropeanMathematicalSociety vol 5 no 1 pp 1ndash40 2003

[26] E Shores ldquoHypoellipticity forlinear degenerate elliptic systemsin Carnot groups and applicationsrdquo httparxivorgabsmath0502569

[27] A Foglein ldquoPartial regularity results for subelliptic systemsin the Heisenberg grouprdquo Calculus of Variations and PartialDifferential Equations vol 32 no 1 pp 25ndash51 2008

[28] G Lu ldquoThe sharp Poincare inequality for free vector fields anendpoint resultrdquo Revista Matematica Iberoamericana vol 10no 2 pp 453ndash466 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 2: Research Article Partial Regularity for Nonlinear Subelliptic Systems …downloads.hindawi.com/journals/aaa/2013/950134.pdf · 2019-07-31 · Partial Regularity for Nonlinear Subelliptic

2 Abstract and Applied Analysis

we also refer the reader to Giaquinta [4] and Chen and Wu[5] for further discussion and additional examples Thenthe goal is to establish partial regularity theory Moreover anew method calledA-harmonic approximation technique isintroduced by Duzaar and Steffen in [6] and simplified byDuzaar and Grotowski in [7] to study elliptic systems withquadratic growth caseThen similar results have been provedfor more general 119860120572

119894or 119861120572 in the Euclidean setting see [8ndash

11] for Holder continuous coefficients and [12ndash15] for Dinicontinuous coefficients

However turning to sub-elliptic equations and systemsin the Heisenberg groups H119899 some new difficulties willarise Already in the first step trying to apply the standarddifference quotient method the main difference betweenEuclidean R119899 and Heisenberg groups H119899 becomes clear Anytime we use horizontal difference quotients (ie in the direc-tions 119883

119894) extra terms with derivatives in the 119879 direction will

arise due to noncommutativity (see (12)) but these cannotbe controlled by using the initial assumptions on the weaksolution Several results were focused on those equationswhich have a bearing on basic vector fields on the Heisenberggroup or more generally the Carnot group Capogna [16 17]studied the regularities for weak solutions to quasi-linearequations Concretely by a technique combining fractionaldifference quotients and fractional derivatives defined byFourier transform differentiability in the nonhorizontaldirection11988222 estimate and119862infin continuity of weak solutionsare obtained see [16] for the case of Heisenberg groups and[17] for Carnot groups To sub-elliptic 119901-Laplace equations inHeisenberg groups Marchi in [18ndash20] showed that 119879119906 isin 119871119901locand 1198832119906 isin 119871

2

loc for 1 + (1radic5) lt 119901 lt 1 +

radic

5 by usingtheories of Besov space and Bessel potential space Domokosin [21 22] improved these results for 1 lt 119901 lt 4 employingthe A Zygmund theory related to vector fields Recently bymeticulous arguments Manfredi and Mingione in [23] andMingione et al in [24] proved Holder regularity with regardto full Euclidean gradient for weak solutions and further 119862infincontinuity under the coefficients assumed to be smooth

While regularities for weak solutions to sub-elliptic sys-tems concerning vector fields aremore complicated Capognaand Garofalo in [25] showed the partial Holder regularityfor the horizontal gradient of weak solutions to quasilinearsub-elliptic systems minussum119896

119894=1119883

119894(119860

120572

119894(120585 119906)119883

119895119906) = 119861

120572

(120585 119906 119883119906)

with 119883119894 119883119895(119894 119895 = 1 119896) being horizontal vector fields

in Carnot groups of step two where 119860120572119894and 119861120572 satisfy the

quadratic structure conditions Their way relies mainly ongeneralization of classical direct method in the Euclideansetting Shores in [26] considered a homogeneous quasi-linear system minussum

119896

119894=1119883

119894(119860

120572

119894(120585 119906)119883

119895119906) = 0 in the Carnot

group with general step where 119860120572119894also satisfies the quadratic

growth condition She established higher differentiability andsmoothness for weak solutions of the system with constantcoefficients and deduced partial regularity for weak solutionsof the original system With respect to the case of non-quadratic growth Foglein in [27] treated the homogeneousnonlinear system minussum

2119899

119894=1119883

119894119860

120572

119894(120585 119883119906) = 0 in the Heisenberg

group under superquadratic structure conditions She got

a priori estimates for weak solutions of the system withconstant coefficients and partial regularity for the horizontalgradient of weak solutions to the initial system Later Wangand Niu [1] and Wang and Liao [2] treated more generalnonlinear sub-elliptic system in the Carnot groups undersuperquadratic growth conditions and subquadratic growthconditions respectively

The regularity results for sub-elliptic systems mentionedabove require Holder continuity with respect to the coef-ficients 119860120572

119894 When the assumption of Holder continuity on

119860

120572

119894is weakened to Dini continuity how to establish partial

regularity of weak solutions to nonlinear sub-elliptic systemsin the Heisenberg group This paper is devoted to this topicTo define weak solution to (1) we assume the followingstructure conditions on 119860120572

119894and 119861120572

(H1) 119860120572119894(120585 119906 119901) is differentiable in 119901 and there exist some

constants 119871 such that1003816

1003816

1003816

1003816

1003816

1003816

1003816

119860

120572

119894119901119895

120573

(120585 119906 119901)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

le 119871 (120585 119906 119901) isin Ω timesR119873

timesR2119899119873

(4)

Here we write down 119860120572119894119901119895

120573

(120585 119906 119901) = (120597119860

120572

119894(120585 119906 119901)

120597119901

119895

120573)

(H2) 119860120572119894(120585 119906 119901) is uniformly elliptic that is for some 120582 gt

0 we have

119860

120572

119894119901119895

120573

(120585 119906 119901) 120578

120572

119894120578

120573

119895ge 120582

1003816

1003816

1003816

1003816

120578

1003816

1003816

1003816

1003816

2

forall120578 isin R2119899119873

(5)

(H3) There exist a modulus of continuity 120583 (0 +infin) rarr

[0 +infin) and a nondecreasing function 120581 [0 +infin) rarr

[1 +infin) such that

(1 +

1003816

1003816

1003816

1003816

119901

1003816

1003816

1003816

1003816

)

minus1 10038161003816

1003816

1003816

1003816

119860

120572

119894(120585 119906 119901) minus 119860

120572

119894(

120585 119901)

1003816

1003816

1003816

1003816

1003816

le 120581 (|119906|) 120583 (119889 (120585

120585) + |119906 minus |)

(6)

(H4) 119861120572 satisfies quadratic controllable growth condition

1003816

1003816

1003816

1003816

119861

120572

(120585 119906 119901)

1003816

1003816

1003816

1003816

le 119862 (1 + |119906|

119903minus1

+

1003816

1003816

1003816

1003816

119901

1003816

1003816

1003816

1003816

2(1minus1119903)

) (7)

where 119903 = 2119876(119876 minus 2) because 119876 gt 2 see (16)

Without loss of generality we can assume that 120581 ge 1 andthe following

(1205831) 120583 is nondecreasing with 120583(0+) = 0(1205832) 120583 is concave in the proof of the regularity theorem

we have to require that 119903 rarr 119903

minus120574

120583(119903) is nonincreasingfor some exponent 120574 isin (0 1) We also require Dinirsquoscondition (2) which was already mentioned in theintroduction

(1205833) 119872(119903) = int1199030

(120583(120588)120588)119889120588 lt infin for some 119903 gt 0

In the present paper we will apply the method of A-harmonic approximation adapting to the setting of Heisen-berg groups to study partial regularity for the system (1) Since

Abstract and Applied Analysis 3

basic vector fields 119883119894of Lie algebras corresponding to the

Heisenberg group are more complicated than gradient vectorfields in the Euclidean setting we have to find a differentauxiliary function in proving Caccioppoli type inequalityBesides the nonhorizontal derivatives of weak solutionswill happen in the Taylor type formula in the Heisenberggroup and cannot be effectively controlled in the presenthypotheses So the method employing Taylorrsquos formula in[12] is not appropriate in our setting In order to obtain thedesired decay estimate we use the Poincare type inequalityin [28] as a replacement And we obtain the following mainresult

Theorem 1 Assume that coefficients 119860120572119894and 119861120572 satisfy (H1)ndash

(H4) (1205831)ndash(1205833) and that 119906 isin 11986711988212(ΩR119873) is a weak solutionto the system (1) that is

int

Ω

119860

120572

119894(120585 119906 119883119906)119883

119894120601

120572

119889120585 = int

Ω

119861

120572

(120585 119906 119883119906) 120601

120572

119889120585

forall120601 isin 119862

infin

0(ΩR

119873

)

(8)

Then there exists a relatively closed set Sing 119906 sub Ω such that119906 isin 119862

1

(Ω Sing 119906 R119873) Furthermore Sing 119906 sub Σ1cup Σ

2and

Haar meas (Ω Sing 119906) = 0 where

Σ

1= 120585

0isin Ω sup

119903gt0

(

1003816

1003816

1003816

1003816

1003816

119906

1205850119903

1003816

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

1003816

(119883119906)

1205850119903

1003816

1003816

1003816

1003816

1003816

) = infin

Σ

2= 120585

0isin Ω lim

119903rarr0+inf 10038161003816

1003816

1003816

119861

119903(120585

0)

1003816

1003816

1003816

1003816

minus1

H119899

timesint

119861119903(1205850)

1003816

1003816

1003816

1003816

1003816

119883119906 minus (119883119906)

1205850119903

1003816

1003816

1003816

1003816

1003816

2

119889120585 gt 0

(9)

In addition for 120591 isin [120574 1) and 1205850isin Ω Sing 119906 the derivative

119883119906 has the modulus of continuity 119903 rarr 119903

120591

+ 119872(119903) in a neigh-borhood of 120585

0

2 Preliminaries

The Heisenberg group H119899 is defined as R2119899+1 endowed withthe following group multiplication

sdot H119899

times H119899

997888rarr H119899

((120585

1

119905) (

120585

1

119905)) 997891997888rarr (120585

1

+

120585

1

119905 +

119905 +

1

2

119899

sum

119894=1

(119909

119894119910

119894minus 119909

119894119910

119894))

(10)

for all 120585 = (120585

1

119905) = (119909

1 119909

2 119909

119899 119910

1 119910

2 119910

119899 119905) 120585 =

(

120585

1

119905) = (119909

1 119909

2 119909

119899 119910

1 119910

2 119910

119899

119905) This multiplicationcorresponds to addition in Euclidean R2119899+1 Its neutralelement is (0 0) and its inverse to (1205851 119905) is given by (minus1205851 minus119905)Particularly the mapping (120585 120585) 997891rarr 120585 sdot

120585

minus1 is smooth so (H119899 sdot)is a Lie group

The basic vector corresponding to its Lie algebra can beexplicitly calculated by the exponential map and is given by

119883

119894=

120597

120597119909

119894

minus

119910

119894

2

120597

120597119905

119883

119894+119899=

120597

120597119910

119894

+

119909

119894

2

120597

120597119905

119879 =

120597

120597119905

(11)

for 119894 = 1 2 119899 and note that the special structure of thecommutators

[119883

119894 119883

119894+119899] = minus [119883

119894+119899 119883

119894] = 119879 else [119883

119894 119883

119895] = 0

[119879 119879] = [119879119883

119894] = 0

(12)

that is (H119899 sdot) is a nilpotent Lie group of step 2 119883 =

(119883

1 119883

2119899) is called the horizontal gradient and 119879 the

vertical derivativeThe pseudonorm is defined by

1003817

1003817

1003817

1003817

1003817

(120585

1

119905)

1003817

1003817

1003817

1003817

1003817

= (

1003816

1003816

1003816

1003816

1003816

120585

11003816

1003816

1003816

1003816

1003816

4

+ 119905

2

)

14

(13)

and the metric induced by this pseudonorm is given by

119889 (

120585 120585) =

1003817

1003817

1003817

1003817

1003817

120585

minus1

sdot

120585

1003817

1003817

1003817

1003817

1003817

(14)

The measure used on H119899 is Haar measure and the volume ofthe pseudoball 119861

119877(120585

0) = 120585 isin H119899 119889(120585

0 120585) lt 119877 is given by

1003816

1003816

1003816

1003816

119861

119877(120585

0)

1003816

1003816

1003816

1003816H119899= 119877

2119899+210038161003816

1003816

1003816

119861

1(120585

0)

1003816

1003816

1003816

1003816H119899≜ 120596

119899119877

2119899+2

(15)

The number

119876 = 2119899 + 2 (16)

is called the homogeneous dimension of H119899The horizontal Sobolev spaces 1198671198821119901(Ω) (1 le 119901 lt infin)

are defined as

119867119882

1119901

(Ω) = 119906 isin 119871

119901

(Ω) 119883

119894119906 isin 119871

119901

(Ω)

119894 = 1 2 2119899

(17)

Then1198671198821119901(Ω) is a Banach space with the norm

119906

1198671198821119901(Ω)

= 119906

119871119901(Ω)+

2119899

sum

119894=1

1003817

1003817

1003817

1003817

119883

119894119906

1003817

1003817

1003817

1003817119871119901(Ω) (18)

119867119882

1119901

0(Ω) is the completion of 119862infin

0(Ω) under norm (18)

Lu [28] showed the following Poincare type inequalityrelated to Hormanderrsquos vector fields for 119906 isin 1198671198821119902(119861

119877(120585

0))

1 lt 119902 lt 119876 1 le 119901 le 119902119876(119876 minus 119902)

(∮

119861119877(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus 119906

1205850119877

1003816

1003816

1003816

1003816

1003816

119901

119889120585)

1119901

le 119862

119901119877(∮

119861119877(1205850)

|119883119906|

119902

119889120585)

1119902

(19)

where we write down ∮119861119903(1205850)

119906119889120585 = |119861

119903(120585

0)|

minus1

H119899 int119861119903(1205850)119906119889120585 here

and there Note the fact that the horizontal vectors119883119894defined

4 Abstract and Applied Analysis

in (11) fit Hormanderrsquos vector fields and that (19) is valid for119901 = 119902 = 2

Following [12] for technical convenience letting 120578(119905) =120583

2

(

radic

2119905) we have the corresponding properties for 120578 (1205781) 120578is continuous nondecreasing and 120578(0) = 0 (1205782) 120578 is concaveand 119903 rarr 119903

minus120574

120578(119903) is nonincreasing for some exponent 120574 isin(0 1) (1205783)119867(119903) = 41198722(radic2119903) = [int119903

0

(

radic

120578(120588)120588)119889120588]

2

lt infin forsome 119903 gt 0 Changing 120581 by a constant but keeping 120581 ge 1 wemay assume the following (1205784) 120578(1) = 1 implying 120578(119905) ge 119905for 119905 isin [0 1] Also note that it implies that from (1205782) and (1205784)120578(119905) le (120574

2

4)119867(119905) for all 119905 ge 0Furthermore the following inequality holds

119904120578 (119905) le 119904120578 (119904) + 119905 119904 isin [0 1] 119905 gt 0 (20)

The condition (H3) becomes

1003816

1003816

1003816

1003816

1003816

119860

120572

119894(120585 119906 119901) minus 119860

120572

119894(

120585 119901)

1003816

1003816

1003816

1003816

1003816

le 120581 (|119906|)radic120578 (119889

2

(120585

120585) + |119906 minus |

2

) (1 +

1003816

1003816

1003816

1003816

119901

1003816

1003816

1003816

1003816

)

(21)

Moreover we deduce the existence of a nonnegative modulusof continuity with 120596(119905 0) = 0 for all 119905 such that 120596(119904 119905) isnondecreasing with respect to 119905 for fixed 119904 and 1205962(119904 119905) isconcave and nondecreasing with respect to 119904 for fixed 119905 Alsowe have for |119906| + |119883119906| le 119872

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119860

120572

119894119901119895

120573

(120585 119906 119901) minus 119860

120572

119894119901119895

120573

(

120585

119901)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

le 120596 (119872 119889

2

(120585

120585) + |119906 minus |

2

+

1003816

1003816

1003816

1003816

119901 minus

119901

1003816

1003816

1003816

1003816

2

)

(22)

Using (H1) and (H2) we see that

1003816

1003816

1003816

1003816

119860

120572

119894(120585 119906 119901) minus 119860

120572

119894(120585 119906

119901)

1003816

1003816

1003816

1003816

le 119871

1003816

1003816

1003816

1003816

119901 minus

119901

1003816

1003816

1003816

1003816

(23)

(119860

120572

119894(120585 119906 119901) minus 119860

120572

119894(120585 119906

119901)) (119901 minus

119901) ge 120582

1003816

1003816

1003816

1003816

119901 minus

119901

1003816

1003816

1003816

1003816

2

(24)

In the sequel the constant 119862may vary from line to line

3 Caccioppoli Type Inequality

In this section we present the followingA-harmonic approx-imation lemma in the Heisenberg group introduced byFoglein [27] with 119901 = 2 as a special case and prove aCaccioppoli type inequality in our setting

Lemma 2 Let 120582 and 119871 be fixed positive numbers and 119899119873 isin

N with 119899 ge 2 If for any given 120576 gt 0 there exists 120575 =

120575(119899119873 120582 120576) isin (0 1] with the following properties

(I) for anyA isin Bil(R2119899119873) satisfying

A (] ]) ge 120582|]|2 A (] ]) le 119871 |]| |]| ] ] isin R2119899119873

(25)

(II) for any 119908 isin 11986711988212(119861120588(120585

0)R119873) satisfying

∮

119861120588(1205850)

|119883119908|

2

119889120585 le 1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

∮

119861120588(1205850)

A (119883119908119883120593) 119889120585

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

le 120575 sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

forall120593 isin 119862

1

0(119861

120588(120585

0) R119873

)

(26)

then there exists anA-harmonic function ℎ such that

∮

119861120588(1205850)

|119883ℎ|

2

119889120585 le 1 120588

minus2

∮

119861120588(1205850)

|ℎ minus 119908|

2

119889120585 le 120576 (27)

Foglein [27] established a priori estimate for the weaksolution 119906 to homogeneous sub-elliptic systemswith constantcoefficients in the Heisenberg group (also see [25] for Carnotgroups of step 2) We list it as follows

sup1198611205882(1205850)

(|119906|

2

+ 120588

2

|119883119906|

2

+ 120588

41003816

1003816

1003816

1003816

1003816

119883

2

119906

1003816

1003816

1003816

1003816

1003816

2

) le 119862

0∮

119861120588(1205850)

|119883119906|

2

119889120585

(28)

In what follows we let 1205881(119904 119905) = (1 + 119904 + 119905)

minus1

120581(119904 + 119905)

minus1 and119870

1(119904 119905) = (1 + 119905)

4

120581(119904 + 119905)

4 for 119904 119905 ge 0 Note that 1205881le 1 and

that 119904 rarr 120588

1(119904 119905) 119905 rarr 120588

1(119904 119905) are nonincreasing functions

Lemma 3 Let 119906 isin 11986711988212(ΩR119873) be a weak solution to thesystem (1) under the conditions (H1)ndash(H4) (1205831)ndash(1205833) Thenfor every 120585

0= (119909

0

1 119909

0

2 119909

0

119899 119910

0

1 119910

0

2 119910

0

119899 119905) isin Ω 119906

0isin R119873

119901

0isin R2119899119873 and 0 lt 120588 lt 119877 lt 120588

1(|119906

0| |119901

0|) le 1 such that

119861

119877(120585

0) subsub Ω the inequality

int

119861120588(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

119889120585

le 119862

119888[

1

(119877 minus 120588)

2int

119861119877(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus 119906

0minus (120585

1

minus 120585

1

0) 119901

0

1003816

1003816

1003816

1003816

1003816

2

119889120585 + 119865]

(29)

holds where 1205851 = (1199091 119909

2 119909

119899 119910

1 119910

2 119910

119899) is the horizon-

tal component of 120585 = (1205851 119905) isin Ω and

119865 = 120596

119899119877

119876

119870

1(

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

) 120578 (119877

2

)

+ [int

119861119877(1205850)

(1 + 119906

119903

+ |119883119906|

2

) 119889120585]

2(1minus1119903)

(30)

Proof Let V = 119906minus1199060minus(120585

1

minus120585

1

0)119901

0 Take a test function120593 = 1206012V

in (8) with 120601 isin 119862infin0(119861

119877(120585

0)R119873) satisfying 0 le 120601 le 1 |nabla120601| le

119862(119877minus120588) and 120601 equiv 1 on 119861120588(120585

0) Then we have119883V = 119883119906minus119901

0

|119883120593| le 120601|119883119906 minus 119901

0| + 119862(119877 minus 120588)|V| and

int

119861119877(1205850)

119860

120572

119894(120585 119906 119883119906) 120601

2

(119883119906 minus 119901

0) 119889120585

= minus2int

119861119877(1205850)

120601119883120601119860

120572

119894(120585 119906 119883119906) V119889120585

+ int

119861119877(1205850)

119861

120572

(120585 119906 119883119906) 120593

120572

119889120585

(31)

Abstract and Applied Analysis 5

Adding this to the equations

minus int

119861119877(1205850)

119860

120572

119894(120585 119906 119901

0) 120601

2

(119883119906 minus 119901

0) 119889120585

= 2int

119861119877(1205850)

120601119883120601119860

120572

119894(120585 119906 119901

0) V119889120585

minus int

119861119877(1205850)

119860

120572

119894(120585 119906 119901

0)119883120593

120572

119889120585

0 = int

119861119877(1205850)

119860

120572

119894(120585

0 119906

0 119901

0)119883120593

120572

(32)

It follows that by using the hypotheses (H1) (H3) (ie (23)(21) resp) and (H4)

int

119861119877(1205850)

[119860

120572

119894(120585 119906 119883119906)

minus119860

120572

119894(120585 119906 119901

0)] 120601

2

(119883119906 minus 119901

0) 119889120585

= 2int

119861119877(1205850)

[119860

120572

119894(120585 119906 119901

0)

minus119860

120572

119894(120585 119906 119883119906)] 120601V119883120601119889120585

+ int

119861119877(1205850)

[119860

120572

119894(120585 119906

0+ (120585

1

minus 120585

1

0) 119901

0 119901

0)

minus119860

120572

119894(120585 119906 119901

0) ]119883120593

120572

119889120585

+ int

119861119877(1205850)

[119860

120572

119894(120585

0 119906

0 119901

0)

minus119860

120572

119894(120585 119906

0+ (120585

1

minus 120585

1

0) 119901

0 119901

0)]119883120593

120572

119889120585

+ int

119861119877(1205850)

119861

120572

(120585 119906 119883119906) 120593

120572

119889120585

le 119868 + 119868119868 + 119868119868119868 + 119868119881 + 119881

(33)

where

119868 = 2119871int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

120601

1003816

1003816

1003816

1003816

|V| 1003816100381610038161003816

119883120601

1003816

1003816

1003816

1003816

119889120585

119868119868 = (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

) 120581 (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+ 119877

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

times int

119861119877(1205850)

radic120578 (|V|2) 100381610038161003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

120601

2

119889120585

119868119868119868 = 2 (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

) 120581 (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+ 119877

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

times int

119861119877(1205850)

radic120578 (|V|2) |V| 100381610038161003816

1003816

119883120601

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

120601

1003816

1003816

1003816

1003816

119889120585

119868119881 = (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

) 120581 (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+ 119877

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

times int

119861119877(1205850)

radic120578 (119877

2

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

2

)) [

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

120601

2

+2

1003816

1003816

1003816

1003816

120601

1003816

1003816

1003816

1003816

|V| 1003816100381610038161003816

119883120601

1003816

1003816

1003816

1003816

] 119889120585

119881 = 119862int

119861119877(1205850)

(1 + |119906|

119903minus1

+ |119883119906|

2(1minus1119903)

) 120593119889120585

(34)

Applying (H2) the left hand side of (33) can be estimated as

120582int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

120601

2

119889120585

le int

119861119877(1205850)

[119860

120572

119894(120585 119906 119883119906) minus 119860

120572

119894(120585 119906 119901

0)] 120601

2

(119883119906 minus 119901

0) 119889120585

(35)

For 120576 gt 0 to be fixed later we have using Youngrsquos inequality

119868 le 120576int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

21003816

1003816

1003816

1003816

120601

1003816

1003816

1003816

1003816

2

119889120585 +

119862119871

2

(119877 minus 120588)

2int

119861119877(1205850)

|V|2119889120585

(36)

Using Jensenrsquos inequality (20) and the fact that 120578(119905119904) le 119905120578(119904)for 119905 ge 1 we arrive at

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

120581

2

(sdot) int

119861119877(1205850)

120578 (|V|2) 119889120585

le 120596

119899119877

119876minus2

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

120581

2

(sdot) 119877

2

120578 (∮

119861119877(1205850)

|V|2119889120585)

le 120596

119899119877

119876minus2

[∮

119861119877(1205850)

|V|2119889120585

+ (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

120581

2

(sdot) 119877

2

120578

times ((1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

120581

2

(sdot) 119877

2

) ]

le 119877

minus2

int

119861119877(1205850)

|V|2119889120585

+ 120596

119899119877

119876

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

4

120581

4

(sdot) 120578 (119877

2

)

(37)

where 120581(sdot) is an abbreviation of the function 120581(|1199060| + |119901

0|)

Also note that the application of (20) in the second lastinequality is possible by our choice 119877 le 120588

1(|119906

0| + |119901

0|)

6 Abstract and Applied Analysis

Using Youngrsquos inequality and (37) in 119868119868 we obtain

119868119868 le 120576int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

21003816

1003816

1003816

1003816

120601

1003816

1003816

1003816

1003816

2

119889120585

+ 120576

minus1

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

120581

2

(sdot) int

119861119877(1205850)

120578 (|V|2) 119889120585

le 120576int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

21003816

1003816

1003816

1003816

120601

1003816

1003816

1003816

1003816

2

119889120585

+

1

120576(119877 minus 120588)

2int

119861119877(1205850)

|V|2119889120585

+ 120576

minus1

120596

119899119877

119876

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

4

120581

4

(sdot) 120578 (119877

2

)

(38)

And similarly we see

119868119868119868 le

4119862

(119877 minus 120588)

2int

119861119877(1205850)

|V|2119889120585

+ (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

120581

2

(sdot) int

119861119877(1205850)

120578 (|V|2) 119889120585

le

119862

(119877 minus 120588)

2int

119861119877(1205850)

|V|2119889120585

+ 120596

119899119877

119876

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

4

120581

4

(sdot) 120578 (119877

2

)

119868119881 le 120576int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

21003816

1003816

1003816

1003816

120601

1003816

1003816

1003816

1003816

2

119889120585

+

4119862120576

(119877 minus 120588)

2int

119861119877(1205850)

|V|2119889120585

+ 120576

minus1

120596

119899119877

119876

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

120581

2

(sdot) 120578

times (∮

119861119877(1205850)

119877

2

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

2

) 119889120585)

le 120576int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

21003816

1003816

1003816

1003816

120601

1003816

1003816

1003816

1003816

2

119889120585

+

119862120576

(119877 minus 120588)

2int

119861119877(1205850)

|V|2119889120585

+ 120576

minus1

120596

119899119877

119876

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

4

120581

4

(sdot) 120578 (119877

2

)

(39)

Here we have used 120581 ge 1 in the last inequality

By Holderrsquos inequality (19) and Youngrsquos inequality onegets

119881 le 119862(int

119861119877(1205850)

1003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119903

119889120585)

1119903

times (int

119861119877(1205850)

(1 + |119906|

119903

+ |119883119906|

2

) 119889120585)

(119903minus1)119903

le 119862(int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

2

119889120585)

12

times (int

119861119877(1205850)

(1 + |119906|

119903

+ |119883119906|

2

) 119889120585)

(119903minus1)119903

le 120576int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

2

119889120585 + 119862 (120576)

times (int

119861119877(1205850)

(1 + |119906|

119903

+ |119883119906|

2

) 119889120585)

2(119903minus1)119903

le 120576int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

21003816

1003816

1003816

1003816

120601

1003816

1003816

1003816

1003816

2

119889120585

+

119862120576

(119877 minus 120588)

2int

119861119877(1205850)

|V|2119889120585

+ 119862 (120576) (int

119861119877(1205850)

(1 + |119906|

119903

+ |119883119906|

2

) 119889120585)

2(119903minus1)119903

(40)

where we have used the fact that |119883120593| le 120601|119883119906 minus 1199010| + 119862(119877 minus

120588)|V|Applying these estimates to (37) we obtain

(120582 minus 4120576) int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

120601

2

119889120585

le

119862 (119871 120576)

(119877 minus 120588)

2int

119861119877(1205850)

|V|2119889120585

+ (120576

minus1

+ 2)120596

119899119877

119876

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

4

120581

4

(sdot) 120578 (119877

2

)

+ 119862 (120576) (int

119861119877(1205850)

(1 + |119906|

119903

+ |119883119906|

2

) 119889120585)

2(119903minus1)119903

(41)

Choosing 120576 = 1205828 we obtain the desired inequality (29)

4 Proof of the Main Theorem

In this section we will complete the proof of the partialregularity results via the following lemmas In the sequel wealways suppose that 119906 isin 11986711988212(ΩR119873) is a weak solution to(1) with the assumptions of (H1)ndash(H4) and (1205831)ndash(1205833)

Abstract and Applied Analysis 7

Lemma 4 Let 119861120588(120585

0) subsub Ω with 120588 le 120588

1(|119906

0| |119901

0|) and 120593 isin

119862

infin

0(119861

120588(120585

0)R119873) satisfying |120593| le 120588

2 and sup119861120588(1205850)

|119883120593| le 1Then there exists a constant 119862

1ge 1 such that

∮

119861120588(1205850)

119860

120572

119894119901119895

120573

(120585

0 119906

0 119901

0) (119883119906 minus 119901

0)119883120593

120572

119889120585

le 119862

1[Φ (120585

0 120588 119901

0)

+ 120596 (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

Φ (120585

0 120588 119901

0))Φ

12

(120585

0 120588 119901

0)

+ 119870

1(

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)radic120578 (120588

2

)] sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

(42)

Proof Using the fact that int119861120588(1205850)

119860

120572

119894(120585

0 119906

0 119901

0)119883120593

120572

119889120585 = 0 andthe weak form (8) we deduce

∮

119861120588(1205850)

[int

1

0

119860

120572

119894119901119895

120573

(120585

0 119906

0 120579119883119906 + (1 minus 120579) 119901

0)

times (119883119906 minus 119901

0) 119889120579]119883120593

120572

119889120585

= ∮

119861120588(1205850)

[119860

120572

119894(120585

0 119906

0 119883119906)

minus119860

120572

119894(120585

0 119906

0 119901

0)]119883120593

120572

119889120585

= ∮

119861120588(1205850)

[119860

120572

119894(120585

0 119906

0 119883119906)

minus119860

120572

119894(120585 119906 119883119906)]119883120593

120572

119889120585

+ ∮

119861120588(1205850)

119861

120572

(120585 119906 119883119906) 120593

120572

119889120585

(43)

It yields

∮

119861120588(1205850)

119860

120572

119894119901119895

120573

(120585

0 119906

0 119901

0) (119883119906 minus 119901

0)119883120593

120572

119889120585

= ∮

119861120588(1205850)

[int

1

0

(119860

120572

119894119901119895

120573

(120585

0 119906

0 119901

0)

minus119860

120572

119894119901119895

120573

(120585

0 119906

0 120579119883119906 + (1 minus 120579) 119901

0))

times (119883119906 minus 119901

0) 119889120579] 119889120585 sup

119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

+ ∮

119861120588(1205850)

[119860

120572

119894(120585

0 119906

0 119883119906)

minus119860

120572

119894(120585 119906

0+ 119901

0(120585 minus 120585

0) 119883119906)] sup

119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

+ ∮

119861120588(1205850)

[119860

120572

119894(120585 119906

0+ 119901

0(120585 minus 120585

0) 119883119906)

minus119860

120572

119894(120585 119906 119883119906)] sup

119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

+ ∮

119861120588(1205850)

119861

120572

(120585 119906 119883119906) 120593

120572

119889120585

= 119868

1015840

+ 119868119868

1015840

+ 119868119868119868

1015840

+ 119868119881

1015840

(44)

Using (22) Holderrsquos inequality the fact that 119905 rarr 120596

2

(119904 119905) isconcave and Jensenrsquos inequality we have

119868

1015840

le sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

∮

119861120588(1205850)

120596 (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

119889120585

le sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

[∮

119861120588(1205850)

120596

2

(

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

) 119889120585]

12

times[∮

119861120588(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

119889120585]

12

le 120596(

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

Φ (120585

0 120588 119901

0))Φ

12

(120585

0 120588 119901

0) sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

(45)

Similarly using (21) and the fact that 120578(119905119904) le 119905120578(119904) for 119905 ge 1we obtain

119868119868

1015840

le sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

120581 (sdot)radic120578 (120588

2

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

2

))

times ∮

119861120588(1205850)

(1 + |119883119906|) 119889120585

le sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

120581 (sdot)radic120578 (120588

2

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

)

times ∮

119861120588(1205850)

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

) 119889120585

le sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

[(∮

119861120588(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

119889120585)

+ 120581

2

(sdot) (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

120578 (120588

2

)

+ 120581 (sdot) (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

3

radic120578 (120588

2

) ]

le [Φ (120585

0 120588 119901

0)+ 2120581

2

(sdot) (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

3

radic120578 (120588

2

)]

times sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

(46)

8 Abstract and Applied Analysis

where we have used the fact that 120578(1205882) le

radic

120578(120588

2

) whichfollows from the nondecreasing property of the function 120578(119905)(1205784) and our assumption 120588 le 120588

1le 1

In the same way it follows that by using (21) (37) and(19)

119868119868119868

1015840

le sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

120581 (sdot)∮

119861120588(1205850)

radic120578 (|V|2) (1 + |119883119906|) 119889120585

le sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

[∮

119861120588(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

119889120585

+ 120581

2

(sdot)∮

119861120588(1205850)

120578 (|V|2) 119889120585

+ 120581 (sdot) (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)∮

119861120588(1205850)

radic120578 (|V|2) 119889120585]

le sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

[Φ (120585

0 120588 119901

0)

+ 2120588

minus2

∮

119861120588(1205850)

|V|2119889120585 + 1205814 (sdot) 120578 (1205882)

+ 120581

2

(sdot) (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

radic120578 (120588

2

) ]

le [(1 + 2119862

119901)Φ (120585

0 120588 119901

0)

+ 2120581

4

(sdot) (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

radic120578 (120588

2

)]

(47)

Using Holderrsquos inequality (19) and Youngrsquos inequality wehave

119868119881

1015840

le 119862∮

119861120588(1205850)

(1 + |119906|

119903minus1

+ |119883119906|

2(1minus1119903)

)

1003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119889120585

le 119862∮

119861120588(1205850)

|119883119906|

2(1minus1119903) 10038161003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119889120585

+ 119862∮

119861120588(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus 119906

0minus 119901

0(120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

119903minus11003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119889120585

+ 119862120588

2

[1 + (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

119903minus1

]

le 119862(∮

119861120588(1205850)

|119883119906|

2

119889120585)

(1minus1119903)

(∮

119861120588(1205850)

1003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119903

119889120585)

1119903

+ 119862(∮

119861120588(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus 119906

0minus 119901

0(120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

119903

119889120585)

(1minus1119903)

times (∮

119861120588(1205850)

1003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119903

119889120585)

1119903

+ 119862120588

2

[1 + (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

119903minus1

]

le 119862(∮

119861120588(1205850)

|119883119906|

2

119889120585)

(1minus1119903)

(∮

119861120588(1205850)

1003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119903

119889120585)

1119903

+ (∮

119861120588(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

119889120585)

(1199032)(1minus1119903)

times (∮

119861120588(1205850)

1003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119903

119889120585)

1119903

+ 119862120588

2

[1 + (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

119903minus1

]

le 119862119906

11986711988212(∮

119861120588(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

119889120585)

(1minus1119903)

times (∮

119861120588(1205850)

1003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119903

119889120585)

1119903

+ 119862120588 [1 + (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

119903minus1

]

le 119862119906

11986711988212(119861120588(1205850))

(∮

119861120588(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

119889120585)

+ 119862120588

2119903

+ 119862120588

2

[1 + (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

119903minus1

]

le 119862

2Φ(120585

0 120588 119901

0) + 119862120588

2

(1 +

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

119903minus1

le 119862

2Φ(120585

0 120588 119901

0) + 119862120581 (sdot) (1 +

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

radic120578 (120588

2

)

(48)

where we have used the assumption (1205784) and the fact that 119903 =2119876(119876 minus 2) = (2119899 + 4)2119899 le 3 and 119862

2= 119862119906

11986711988212(119861120588(1205850))

ge

1 Combining these estimates we obtain the conclusion with119862

1= (1 + 119862

2+ 2119862

119901) ge 1

Lemma 5 Assume that the conditions of Lemma 2 and thefollowing smallness conditions hold

120596 (

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

1003816

(119883119906)

1205850120588

1003816

1003816

1003816

1003816

1003816

Φ (120585

0 120588 (119883119906)

1205850120588))

+ Φ

12

(120585

0 120588 (119883119906)

1205850120588) le

120575

2

(49)

119862

3119870

2

1(

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

1205850120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

) le 120575

2 (50)

with 1198623= 8119862

2

1119862

5 together with

120588 le 120588

1(1 +

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

1 +

1003816

1003816

1003816

1003816

1003816

(119883119906)

1205850120588

1003816

1003816

1003816

1003816

1003816

) (51)

Then the following growth condition holds for 120591 isin [120574 1)

Φ (120585

0 120579120588) le 120579

2120591

Φ(120585

0 120588)

+ 119870

lowast

(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

)

(52)

where one abbreviates Φ(1205850 119903) = Φ(120585

0 119903 (119883119906)

1205850119903) and

119870

lowast

(119904 119905) = 119870(119904 119905) + (2 + 119904 + 119905)

2(119903minus1) with 119870(119904 119905) = (4120575

minus2

+

2

119876

119862

119888)119870

2

1(1 + 119904 1 + 119905)

Proof We define 119908 = [119906 minus 1199061205850120588

minus 119901

0(120585

1

minus 120585

1

0)]120590

minus1 where

120590 = 119862

1radic

Φ(120585

0 120588 119901

0) + 4120575

minus2119870

2

1(

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

) 120578 (120588

2)

(53)

Abstract and Applied Analysis 9

Then we have 119883119908 = 120590

minus1

(119883119906 minus 119901

0) Now we consider

119861

120588(120585

0) subsub Ω such that 120588 le 120588

1(|119906

0| |119901

0|) Applying Lemma 4

on 119861120588(120585

0) to 119906 we have for any 120593 isin 119862infin

0(119861

120588(120585

0)R119873)

∮

119861120588(1205850)

|119883119908|

2

119889120585 = 120590

minus2

Φ(120585

0 120588 119901

0) le

1

119862

2

1

le 1 (54)

∮

119861120588(1205850)

119860

120572

119894119901119895

120573

(120585

0 119906

0 119901

0)119883119908119883120593119889120585

le [Φ

12

(120585

0 120588 119901

0)

+120596 (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

Φ (120585

0 120588 119901

0)) +

120575

2

] sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

(55)

In consideration of the small condition (49) we see that(54) and (55) imply conditions (26) in Lemma 2 Also notethat (H1) and (H3) imply condition (25) So there exists an119860

120572

119894119901119895

120573

(120585

0 119906

0 119901

0)-harmonic function ℎ isin 11986711988212(119861

120588(120585

0)R119873)

such that

∮

119861120588(1205850)

|119883ℎ|

2

119889120585 le 1 120588

minus2

∮

119861120588(1205850)

|119908 minus ℎ|

2

119889120585 le 120576 (56)

Taking 1199060= 119906

12058502120579120588 120579 isin (0 14] and replacing 119901

0by 1199010+

120590(119883ℎ)

1205850 2120579120588 we use Lemma 3 to obtain

int

119861120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119883119906 minus 119901

0minus 120590(119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

2

119889120585

le 119862

119888[

1

(120579120588)

2int

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus 119906

12058502120579120588

minus (119901

0+ 120590(119883ℎ)

12058502120579120588)

times (120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

2

119889120585 + 119865]

(57)

where

119865 = 120596

119899(2120579120588)

119876

119870

1(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119901

0+ 120590(119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 ((2120579120588)

2

)

+ [int

1198612120579120588(1205850)

(1 + 119906

119903

+ |119883119906|

2

) 119889120585]

2(1minus1119903)

(58)

Using the fact that 119906 minus (1199010+ 120590(119883ℎ)

12058502120579120588)(120585

1

minus 120585

1

0) has mean

value 11990612058502120579120588

on the ball 1198612120579120588(120585

0) the definition of119908 and (19)

we have

1

(120579120588)

2∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus 119906

1205850 2120579120588minus (119901

0+ 120590(119883ℎ)

12058502120579120588) (120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

2

119889120585

le

4120590

2

(2120579120588)

2∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119908 minus ℎ

1205850 2120579120588

minus(119883ℎ)

12058502120579120588(120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

2

119889120585

le

4120590

2

(2120579120588)

2[∮

1198612120579120588(1205850)

|119908 minus ℎ|

2

119889120585

+ ∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

ℎ minus ℎ

1205850 2120579120588

minus(119883ℎ)

12058502120579120588(120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

2

119889120585]

le 4120590

2

[(2120579)

minus119876minus2

120576 + 119862

119901∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119883ℎ minus (119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

2

119889120585]

le 4120590

2

[(2120579)

minus119876minus2

120576 + 119862

2

119901(2120579120588)

2

∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119883

2

ℎ

1003816

1003816

1003816

1003816

1003816

2

119889120585]

le 4120590

2

[(2120579)

minus119876minus2

120576 + 119862

2

119901(2120579)

2

119862

0]

le 119862

4(120579

minus119876minus2

120576 + 120579

2

) [Φ (120585

0 120588 119901

0)

+ 4120575

minus2

119870

2

1(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

) 120578 (120588

2

)]

(59)

where 1198624= 119862

4(119876 120582 119871) ge 1 Note that in the second last

inequality we have used the fact that

∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119883

2

ℎ

1003816

1003816

1003816

1003816

1003816

119889120585 le sup119861120588(1205850)

1003816

1003816

1003816

1003816

1003816

119883

2

ℎ

1003816

1003816

1003816

1003816

1003816

le 119862

0120588

minus2

∮

119861120588(1205850)

|119883ℎ|

2

119889120585 le 119862

0120588

minus2

(60)

In consideration of the fact that 119903 = 2119876(119876 minus 2) gt 2 119876 ge 4

and the assumptions 120579 isin (0 14] and Φ le 1 it follows that

[∮

1198612120579120588(1205850)

(1 + 119906

119903

+ |119883119906|

2

) 119889120585]

2(1minus1119903)

le 119862[∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

119889120585]

2(1minus1119903)

+ 119862(∮

1198612120579120588(1205850)

|119883119906|

2

119889120585)

119903minus1

+ (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

4(1minus1119903)

)

10 Abstract and Applied Analysis

le 119862 [(2120579)

minus2119876(1minus1119903)

Φ(120585

0 120588 119901

0)

2(1minus1119903)

+(2120579)

minus119876(119903minus1)

Φ(120585

0 120588 119901

0)

119903minus1

]

+ (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

2(1minus1119903)

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

4(1minus1119903)

)

le 119862(2120579)

minus119876(119903minus1)

Φ(120585

0 120588 119901

0)

2(1minus1119903)

+ (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

2(1minus1119903)

)

2

(61)

Let 119875 = 1199010+120590(119883ℎ)

12058502120579120588with 119901

0= (119883119906)

12058502120579120588 Combining

these estimates (57)ndash(61) and considering the small condition(51) (it implies 120588 le 120588

1(|119906

12058502120579120588| |119875|) see (64) and (65)) we

deduce that

Φ(120585

0 120579120588) le

1003816

1003816

1003816

1003816

1003816

119861

120579120588(120585

0)

1003816

1003816

1003816

1003816

1003816

minus1

H119899int

1198612120579120588(1205850)

|119883119906 minus 119875|

2

119889120585

le 119862

119888

2

119876

(120579120588)

2∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus 119906

12058502120579120588

minus (119901

0+ 120590(119883ℎ)

12058502120579120588)

times (120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

2

119889120585

+ 2

119876

119862

119888119870

1(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119901

0+ 120590(119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

)

+ 119862

119888

(2120579120588)

2119876(1minus1119903)

(120579120588)

119876

times[∮

1198612120579120588(1205850)

(1 + 119906

119903

+ |119883119906|

2

) 119889120585]

2(1minus1119903)

le 2

119876

119862

4119862

119888(120579

minus119876minus2

120576 + 120579

2

)

times [Φ (120585

0 120588) + 4120575

minus2

119870

2

1

times (

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

) ]

+ 2

119876

119862

119888119870

1(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

+120590(119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

)

+ [2

119876

119862

119888(2120579)

2minus119876(119903minus1)

Φ(120585

0 120588)

2(1minus1119903)

+(1 +

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

2(1minus1119903)

)

2

] 120588

2

(62)

Wenow specify 120576 = 120579119876+4 120579 isin (0 14] such that 2119876+11198624119862

119888120579

2

le

120579

2120591 Note that the small condition (50) implies 12059021198625le 1 with

119862

5= max119862

0 119862

1198882

119876

(2120579)

minus(1198762+4)(119876minus2)

and it yields

2

119876

119862

119888(2120579)

2minus119876(119903minus1)

Φ(120585

0 120588)

2(1minus1119903)

le 1

(63)1003816

1003816

1003816

1003816

1003816

120590(119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

le 120590 sup1198612120579120588(1205850)

|119883ℎ|

le 120590radic119862

0(∮

119861120588(1205850)

|119883ℎ|

2

119889120585) le 120590radic119862

0le 1

(64)

where we have used the a priori estimate (28) for the A-harmonic function ℎ Furthermore using (19) and recallingthe definition of 120590 and 119862

1 we have

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

le

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588minus 119906

1205850120588

1003816

1003816

1003816

1003816

1003816

le

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+ (2120579)

minus1198762

times (∮

119861120588(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus (119883119906)

1205850120588(120585

1

minus 120585

1

0) minus 119906

1205850120588

1003816

1003816

1003816

1003816

1003816

2

119889120585)

12

le

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+ (2120579)

minus1198762

120588radic119862

119901Φ

12

(120585

0 120588)

le

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+

120590

radic

119862

119901

119862

1(2120579)

1198762

le

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+ 120590radic119862

5le

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+ 1

(65)

Combining these estimates with (62) we have

Φ(120585

0 120579120588) le 120579

2120591

Φ(120585

0 120588)

+ [4120575

minus2

119870

2

1(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

)

+ 2

119876

119862

119888119870

1(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

+120590(119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

)] 120578 (120588

2

)

+ [1 + (1 +

1003816

1003816

1003816

1003816

1003816

(119883119906)

1205850 2120579120588

1003816

1003816

1003816

1003816

1003816

2(1minus1119903)

)

2

] 120578 (120588

2

)

le 120579

2120591

Φ(120585

0 120588) + 119870 (

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

)

+ (2 +

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

)

2(119903minus1)

120578 (120588

2

)

le 120579

2120591

Φ(120585

0 120588) + 119870

lowast

(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

)

(66)

Then the proof of Lemma 5 is complete

Abstract and Applied Analysis 11

For 119879 gt 0 we find Φ0(119879) gt 0 (depending on 119876119873 120582 119871

120591 and 120596) such that

120596

2

(2119879 2Φ

0(119879)) + 2Φ

0(119879) le

1

2

120575

2

119862

1Φ

0(119879) le 120579

119876

(1 minus 120579

120591

)

2

119879

2

(67)

With Φ0(119879) from (67) we choose 120588

0(119879) isin (0 1] (depending

on 119876119873 120582 119871 120591 120596 120578 and 120581) such that

120588

0(119879) le 120588

1(1 + 2119879 1 + 2119879)

119862

3119870

2

1(2119879 2119879) 120578 (120588

0(119879)

2

) le 120575

2

119870

0(119879) 120578 (120588

0(119879)

2

) le (120579

2120574

minus 120579

2120591

)Φ

0(119879)

2 (1 + 119862

119901)119870

0(119879)119867 (120588

0(119879)

2

) le 120579

119876

(1 minus 120579

120574

)

2

(120579

2120574

minus 120579

2120591

) 119879

2

(68)

where1198700(119879) = 119870

lowast

(2119879 2119879)By the proof method of of Lemma 51 in [12] and

conditions (67)-(68) Lemma 6 can be proved As is wellknown it is sufficient to complete the proof ofTheorem 1 oncewe obtain Lemma 6

Lemma 6 Assume that for some 1198790gt 0 and 119861

120588(120585

0) subsub Ω one

has

(1) |1199061205850120588| + |(119883119906)

1205850120588| le 119879

0

(2) 120588 le 1205880(119879

0)

(3) Φ(1205850 120588) le Φ

0(119879

0)

Then the small conditions (49)ndash(51) are satisfied on the balls119861

120579119895120588(120585

0) for 119895 isin 119873 cup 0 Moreover the limit Λ

1205850=

lim119895rarrinfin

(119883119906)

1205850120579119895120588exists and the inequality

∮

119861120588(1205850)

1003816

1003816

1003816

1003816

1003816

119883119906 minus Λ

1205850

1003816

1003816

1003816

1003816

1003816

2

119889120585 le 119862

6((

119903

120588

)

2120591

Φ(120585

0 120588) + 119867 (119903

2

))

(69)

is valid for 0 lt 119903 le 120588 with a constant 1198626= 119862

6(119876 119873 120582 119871

120591 119886119899119889 119879

0)

Proof The proof is very similar to the proof of Lemma 51 in[12] We omit it here

Acknowledgments

The project was supported by the National Natural ScienceFoundation of China (no 11201081 and no 11126294) andby the Science and Technology Planning Project of JiangxiProvince China no GJJ13657

References

[1] J Wang and P Niu ldquoOptimal partial regularity for weaksolutions of nonlinear sub-elliptic systems in Carnot groupsrdquoNonlinear Analysis Theory Methods amp Applications vol 72 no11 pp 4162ndash4187 2010

[2] J Wang and D Liao ldquoOptimal partial regularity for sub-elliptic systems with sub-quadratic growth in Carnot groupsrdquoNonlinear Analysis Theory Methods amp Applications vol 75 no4 pp 2499ndash2519 2012

[3] E de Giorgi ldquoUn esempio di estremali discontinue per unproblema variazionale di tipo ellitticordquo Bollettino della UnioneMatematica Italiana vol 4 pp 135ndash137 1968

[4] M Giaquinta Multiple Integrals in the Calculus of Variationsand Nonlinear Elliptic Systems Princeton University PressPrinceton NJ USA 1983

[5] Y Chen and L Wu Second Order Elliptic Equations and EllipticSystems Science Press Beijing China 2003

[6] F Duzaar and K Steffen ldquoOptimal interior and boundary reg-ularity for almost minimizers to elliptic variational integralsrdquoJournal fur die Reine und Angewandte Mathematik vol 546 pp73ndash138 2002

[7] F Duzaar and J F Grotowski ldquoOptimal interior partial regu-larity for nonlinear elliptic systems the method of A-harmonicapproximationrdquo Manuscripta Mathematica vol 103 no 3 pp267ndash298 2000

[8] F Duzaar J F Grotowski and M Kronz ldquoRegularity ofalmost minimizers of quasi-convex variational integrals withsubquadratic growthrdquo Annali di Matematica Pura ed ApplicataIV vol 184 no 4 pp 421ndash448 2005

[9] F Duzaar and G Mingione ldquoThe p-harmonic approximationand the regularity of p-harmonic mapsrdquo Calculus of Variationsand Partial Differential Equations vol 20 no 3 pp 235ndash2562004

[10] F Duzaar and G Mingione ldquoRegularity for degenerate ellipticproblems via p-harmonic approximationrdquo Annales de lrsquoInstitutHenri Poincare Analyse Non Lineaire vol 21 no 5 pp 735ndash7662004

[11] S Chen andZ Tan ldquoThemethod ofA-harmonic approximationand optimal interior partial regularity for nonlinear ellipticsystems under the controllable growth conditionrdquo Journal ofMathematical Analysis and Applications vol 335 no 1 pp 20ndash42 2007

[12] F Duzaar and A Gastel ldquoNonlinear elliptic systems with Dinicontinuous coefficientsrdquo Archiv der Mathematik vol 78 no 1pp 58ndash73 2002

[13] F Duzaar A Gastel and G Mingione ldquoElliptic systemssingular sets and Dini continuityrdquo Communications in PartialDifferential Equations vol 29 no 7-8 pp 1215ndash1240 2004

[14] Y Qiu and Z Tan ldquoOptimal interior partial regularity fornonlinear elliptic systems with Dini continuous coefficientsrdquoActa Mathematica Scientia B vol 30 no 5 pp 1541ndash1554 2010

[15] Y Qiu ldquoOptimal partial regularity of second order nonlin-ear elliptic systems with Dini continuous coefficients for thesuperquadratic caserdquo Nonlinear Analysis Theory Methods ampApplications vol 75 no 8 pp 3574ndash3590 2012

[16] L Capogna ldquoRegularity of quasi-linear equations in theHeisen-berg grouprdquoCommunications on Pure andAppliedMathematicsvol 50 no 9 pp 867ndash889 1997

[17] L Capogna ldquoRegularity for quasilinear equations and 1-quasiconformal maps in Carnot groupsrdquo MathematischeAnnalen vol 313 no 2 pp 263ndash295 1999

[18] S Marchi ldquo1198621120572 local regularity for the solutions of the p-Laplacian on the Heisenberg group for 2 lt 119901 lt 1 +

radic

5rdquoZeitschrift fur Analysis und ihre Anwendungen vol 20 no 3 pp617ndash636 2001

12 Abstract and Applied Analysis

[19] S Marchi ldquo1198621120572 local regularity for the solutions of the p-Laplacian on the Heisenberg groupThe case 1+1radic5 lt 119901 le 2rdquoCommentationes Mathematicae Universitatis Carolinae vol 44no 1 pp 33ndash56 2003

[20] S Marchi ldquoL119901 regularity of the derivative in the secondcommutator direction for nonlinear elliptic equations on theHeisenberg grouprdquo Accademia Nazionale delle Scienze detta deiXL Rendiconti Serie V Memorie di Matematica e ApplicazioniParte I vol 26 pp 1ndash15 2002

[21] A Domokos ldquoDifferentiability of solutions for the non-degenerate p-Laplacian in the Heisenberg grouprdquo Journal ofDifferential Equations vol 204 no 2 pp 439ndash470 2004

[22] A Domokos On the regularity of p-harmonic functions inthe Heisenberg group [PhD thesis] University of PittsburghPittsburgh Pa USA 2004

[23] J J Manfredi and G Mingione ldquoRegularity results for quasilin-ear elliptic equations in the Heisenberg grouprdquo MathematischeAnnalen vol 339 no 3 pp 485ndash544 2007

[24] G Mingione A Zatorska-Goldstein and X Zhong ldquoGradientregularity for elliptic equations in the Heisenberg grouprdquoAdvances in Mathematics vol 222 no 1 pp 62ndash129 2009

[25] L Capogna and N Garofalo ldquoRegularity of minimizers of thecalculus of variations inCarnot groups via hypoellipticity of sys-tems ofHormander typerdquo Journal of the EuropeanMathematicalSociety vol 5 no 1 pp 1ndash40 2003

[26] E Shores ldquoHypoellipticity forlinear degenerate elliptic systemsin Carnot groups and applicationsrdquo httparxivorgabsmath0502569

[27] A Foglein ldquoPartial regularity results for subelliptic systemsin the Heisenberg grouprdquo Calculus of Variations and PartialDifferential Equations vol 32 no 1 pp 25ndash51 2008

[28] G Lu ldquoThe sharp Poincare inequality for free vector fields anendpoint resultrdquo Revista Matematica Iberoamericana vol 10no 2 pp 453ndash466 1994

Submit your manuscripts athttpwwwhindawicom

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 3: Research Article Partial Regularity for Nonlinear Subelliptic Systems …downloads.hindawi.com/journals/aaa/2013/950134.pdf · 2019-07-31 · Partial Regularity for Nonlinear Subelliptic

Abstract and Applied Analysis 3

basic vector fields 119883119894of Lie algebras corresponding to the

Heisenberg group are more complicated than gradient vectorfields in the Euclidean setting we have to find a differentauxiliary function in proving Caccioppoli type inequalityBesides the nonhorizontal derivatives of weak solutionswill happen in the Taylor type formula in the Heisenberggroup and cannot be effectively controlled in the presenthypotheses So the method employing Taylorrsquos formula in[12] is not appropriate in our setting In order to obtain thedesired decay estimate we use the Poincare type inequalityin [28] as a replacement And we obtain the following mainresult

Theorem 1 Assume that coefficients 119860120572119894and 119861120572 satisfy (H1)ndash

(H4) (1205831)ndash(1205833) and that 119906 isin 11986711988212(ΩR119873) is a weak solutionto the system (1) that is

int

Ω

119860

120572

119894(120585 119906 119883119906)119883

119894120601

120572

119889120585 = int

Ω

119861

120572

(120585 119906 119883119906) 120601

120572

119889120585

forall120601 isin 119862

infin

0(ΩR

119873

)

(8)

Then there exists a relatively closed set Sing 119906 sub Ω such that119906 isin 119862

1

(Ω Sing 119906 R119873) Furthermore Sing 119906 sub Σ1cup Σ

2and

Haar meas (Ω Sing 119906) = 0 where

Σ

1= 120585

0isin Ω sup

119903gt0

(

1003816

1003816

1003816

1003816

1003816

119906

1205850119903

1003816

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

1003816

(119883119906)

1205850119903

1003816

1003816

1003816

1003816

1003816

) = infin

Σ

2= 120585

0isin Ω lim

119903rarr0+inf 10038161003816

1003816

1003816

119861

119903(120585

0)

1003816

1003816

1003816

1003816

minus1

H119899

timesint

119861119903(1205850)

1003816

1003816

1003816

1003816

1003816

119883119906 minus (119883119906)

1205850119903

1003816

1003816

1003816

1003816

1003816

2

119889120585 gt 0

(9)

In addition for 120591 isin [120574 1) and 1205850isin Ω Sing 119906 the derivative

119883119906 has the modulus of continuity 119903 rarr 119903

120591

+ 119872(119903) in a neigh-borhood of 120585

0

2 Preliminaries

The Heisenberg group H119899 is defined as R2119899+1 endowed withthe following group multiplication

sdot H119899

times H119899

997888rarr H119899

((120585

1

119905) (

120585

1

119905)) 997891997888rarr (120585

1

+

120585

1

119905 +

119905 +

1

2

119899

sum

119894=1

(119909

119894119910

119894minus 119909

119894119910

119894))

(10)

for all 120585 = (120585

1

119905) = (119909

1 119909

2 119909

119899 119910

1 119910

2 119910

119899 119905) 120585 =

(

120585

1

119905) = (119909

1 119909

2 119909

119899 119910

1 119910

2 119910

119899

119905) This multiplicationcorresponds to addition in Euclidean R2119899+1 Its neutralelement is (0 0) and its inverse to (1205851 119905) is given by (minus1205851 minus119905)Particularly the mapping (120585 120585) 997891rarr 120585 sdot

120585

minus1 is smooth so (H119899 sdot)is a Lie group

The basic vector corresponding to its Lie algebra can beexplicitly calculated by the exponential map and is given by

119883

119894=

120597

120597119909

119894

minus

119910

119894

2

120597

120597119905

119883

119894+119899=

120597

120597119910

119894

+

119909

119894

2

120597

120597119905

119879 =

120597

120597119905

(11)

for 119894 = 1 2 119899 and note that the special structure of thecommutators

[119883

119894 119883

119894+119899] = minus [119883

119894+119899 119883

119894] = 119879 else [119883

119894 119883

119895] = 0

[119879 119879] = [119879119883

119894] = 0

(12)

that is (H119899 sdot) is a nilpotent Lie group of step 2 119883 =

(119883

1 119883

2119899) is called the horizontal gradient and 119879 the

vertical derivativeThe pseudonorm is defined by

1003817

1003817

1003817

1003817

1003817

(120585

1

119905)

1003817

1003817

1003817

1003817

1003817

= (

1003816

1003816

1003816

1003816

1003816

120585

11003816

1003816

1003816

1003816

1003816

4

+ 119905

2

)

14

(13)

and the metric induced by this pseudonorm is given by

119889 (

120585 120585) =

1003817

1003817

1003817

1003817

1003817

120585

minus1

sdot

120585

1003817

1003817

1003817

1003817

1003817

(14)

The measure used on H119899 is Haar measure and the volume ofthe pseudoball 119861

119877(120585

0) = 120585 isin H119899 119889(120585

0 120585) lt 119877 is given by

1003816

1003816

1003816

1003816

119861

119877(120585

0)

1003816

1003816

1003816

1003816H119899= 119877

2119899+210038161003816

1003816

1003816

119861

1(120585

0)

1003816

1003816

1003816

1003816H119899≜ 120596

119899119877

2119899+2

(15)

The number

119876 = 2119899 + 2 (16)

is called the homogeneous dimension of H119899The horizontal Sobolev spaces 1198671198821119901(Ω) (1 le 119901 lt infin)

are defined as

119867119882

1119901

(Ω) = 119906 isin 119871

119901

(Ω) 119883

119894119906 isin 119871

119901

(Ω)

119894 = 1 2 2119899

(17)

Then1198671198821119901(Ω) is a Banach space with the norm

119906

1198671198821119901(Ω)

= 119906

119871119901(Ω)+

2119899

sum

119894=1

1003817

1003817

1003817

1003817

119883

119894119906

1003817

1003817

1003817

1003817119871119901(Ω) (18)

119867119882

1119901

0(Ω) is the completion of 119862infin

0(Ω) under norm (18)

Lu [28] showed the following Poincare type inequalityrelated to Hormanderrsquos vector fields for 119906 isin 1198671198821119902(119861

119877(120585

0))

1 lt 119902 lt 119876 1 le 119901 le 119902119876(119876 minus 119902)

(∮

119861119877(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus 119906

1205850119877

1003816

1003816

1003816

1003816

1003816

119901

119889120585)

1119901

le 119862

119901119877(∮

119861119877(1205850)

|119883119906|

119902

119889120585)

1119902

(19)

where we write down ∮119861119903(1205850)

119906119889120585 = |119861

119903(120585

0)|

minus1

H119899 int119861119903(1205850)119906119889120585 here

and there Note the fact that the horizontal vectors119883119894defined

4 Abstract and Applied Analysis

in (11) fit Hormanderrsquos vector fields and that (19) is valid for119901 = 119902 = 2

Following [12] for technical convenience letting 120578(119905) =120583

2

(

radic

2119905) we have the corresponding properties for 120578 (1205781) 120578is continuous nondecreasing and 120578(0) = 0 (1205782) 120578 is concaveand 119903 rarr 119903

minus120574

120578(119903) is nonincreasing for some exponent 120574 isin(0 1) (1205783)119867(119903) = 41198722(radic2119903) = [int119903

0

(

radic

120578(120588)120588)119889120588]

2

lt infin forsome 119903 gt 0 Changing 120581 by a constant but keeping 120581 ge 1 wemay assume the following (1205784) 120578(1) = 1 implying 120578(119905) ge 119905for 119905 isin [0 1] Also note that it implies that from (1205782) and (1205784)120578(119905) le (120574

2

4)119867(119905) for all 119905 ge 0Furthermore the following inequality holds

119904120578 (119905) le 119904120578 (119904) + 119905 119904 isin [0 1] 119905 gt 0 (20)

The condition (H3) becomes

1003816

1003816

1003816

1003816

1003816

119860

120572

119894(120585 119906 119901) minus 119860

120572

119894(

120585 119901)

1003816

1003816

1003816

1003816

1003816

le 120581 (|119906|)radic120578 (119889

2

(120585

120585) + |119906 minus |

2

) (1 +

1003816

1003816

1003816

1003816

119901

1003816

1003816

1003816

1003816

)

(21)

Moreover we deduce the existence of a nonnegative modulusof continuity with 120596(119905 0) = 0 for all 119905 such that 120596(119904 119905) isnondecreasing with respect to 119905 for fixed 119904 and 1205962(119904 119905) isconcave and nondecreasing with respect to 119904 for fixed 119905 Alsowe have for |119906| + |119883119906| le 119872

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119860

120572

119894119901119895

120573

(120585 119906 119901) minus 119860

120572

119894119901119895

120573

(

120585

119901)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

le 120596 (119872 119889

2

(120585

120585) + |119906 minus |

2

+

1003816

1003816

1003816

1003816

119901 minus

119901

1003816

1003816

1003816

1003816

2

)

(22)

Using (H1) and (H2) we see that

1003816

1003816

1003816

1003816

119860

120572

119894(120585 119906 119901) minus 119860

120572

119894(120585 119906

119901)

1003816

1003816

1003816

1003816

le 119871

1003816

1003816

1003816

1003816

119901 minus

119901

1003816

1003816

1003816

1003816

(23)

(119860

120572

119894(120585 119906 119901) minus 119860

120572

119894(120585 119906

119901)) (119901 minus

119901) ge 120582

1003816

1003816

1003816

1003816

119901 minus

119901

1003816

1003816

1003816

1003816

2

(24)

In the sequel the constant 119862may vary from line to line

3 Caccioppoli Type Inequality

In this section we present the followingA-harmonic approx-imation lemma in the Heisenberg group introduced byFoglein [27] with 119901 = 2 as a special case and prove aCaccioppoli type inequality in our setting

Lemma 2 Let 120582 and 119871 be fixed positive numbers and 119899119873 isin

N with 119899 ge 2 If for any given 120576 gt 0 there exists 120575 =

120575(119899119873 120582 120576) isin (0 1] with the following properties

(I) for anyA isin Bil(R2119899119873) satisfying

A (] ]) ge 120582|]|2 A (] ]) le 119871 |]| |]| ] ] isin R2119899119873

(25)

(II) for any 119908 isin 11986711988212(119861120588(120585

0)R119873) satisfying

∮

119861120588(1205850)

|119883119908|

2

119889120585 le 1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

∮

119861120588(1205850)

A (119883119908119883120593) 119889120585

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

le 120575 sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

forall120593 isin 119862

1

0(119861

120588(120585

0) R119873

)

(26)

then there exists anA-harmonic function ℎ such that

∮

119861120588(1205850)

|119883ℎ|

2

119889120585 le 1 120588

minus2

∮

119861120588(1205850)

|ℎ minus 119908|

2

119889120585 le 120576 (27)

Foglein [27] established a priori estimate for the weaksolution 119906 to homogeneous sub-elliptic systemswith constantcoefficients in the Heisenberg group (also see [25] for Carnotgroups of step 2) We list it as follows

sup1198611205882(1205850)

(|119906|

2

+ 120588

2

|119883119906|

2

+ 120588

41003816

1003816

1003816

1003816

1003816

119883

2

119906

1003816

1003816

1003816

1003816

1003816

2

) le 119862

0∮

119861120588(1205850)

|119883119906|

2

119889120585

(28)

In what follows we let 1205881(119904 119905) = (1 + 119904 + 119905)

minus1

120581(119904 + 119905)

minus1 and119870

1(119904 119905) = (1 + 119905)

4

120581(119904 + 119905)

4 for 119904 119905 ge 0 Note that 1205881le 1 and

that 119904 rarr 120588

1(119904 119905) 119905 rarr 120588

1(119904 119905) are nonincreasing functions

Lemma 3 Let 119906 isin 11986711988212(ΩR119873) be a weak solution to thesystem (1) under the conditions (H1)ndash(H4) (1205831)ndash(1205833) Thenfor every 120585

0= (119909

0

1 119909

0

2 119909

0

119899 119910

0

1 119910

0

2 119910

0

119899 119905) isin Ω 119906

0isin R119873

119901

0isin R2119899119873 and 0 lt 120588 lt 119877 lt 120588

1(|119906

0| |119901

0|) le 1 such that

119861

119877(120585

0) subsub Ω the inequality

int

119861120588(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

119889120585

le 119862

119888[

1

(119877 minus 120588)

2int

119861119877(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus 119906

0minus (120585

1

minus 120585

1

0) 119901

0

1003816

1003816

1003816

1003816

1003816

2

119889120585 + 119865]

(29)

holds where 1205851 = (1199091 119909

2 119909

119899 119910

1 119910

2 119910

119899) is the horizon-

tal component of 120585 = (1205851 119905) isin Ω and

119865 = 120596

119899119877

119876

119870

1(

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

) 120578 (119877

2

)

+ [int

119861119877(1205850)

(1 + 119906

119903

+ |119883119906|

2

) 119889120585]

2(1minus1119903)

(30)

Proof Let V = 119906minus1199060minus(120585

1

minus120585

1

0)119901

0 Take a test function120593 = 1206012V

in (8) with 120601 isin 119862infin0(119861

119877(120585

0)R119873) satisfying 0 le 120601 le 1 |nabla120601| le

119862(119877minus120588) and 120601 equiv 1 on 119861120588(120585

0) Then we have119883V = 119883119906minus119901

0

|119883120593| le 120601|119883119906 minus 119901

0| + 119862(119877 minus 120588)|V| and

int

119861119877(1205850)

119860

120572

119894(120585 119906 119883119906) 120601

2

(119883119906 minus 119901

0) 119889120585

= minus2int

119861119877(1205850)

120601119883120601119860

120572

119894(120585 119906 119883119906) V119889120585

+ int

119861119877(1205850)

119861

120572

(120585 119906 119883119906) 120593

120572

119889120585

(31)

Abstract and Applied Analysis 5

Adding this to the equations

minus int

119861119877(1205850)

119860

120572

119894(120585 119906 119901

0) 120601

2

(119883119906 minus 119901

0) 119889120585

= 2int

119861119877(1205850)

120601119883120601119860

120572

119894(120585 119906 119901

0) V119889120585

minus int

119861119877(1205850)

119860

120572

119894(120585 119906 119901

0)119883120593

120572

119889120585

0 = int

119861119877(1205850)

119860

120572

119894(120585

0 119906

0 119901

0)119883120593

120572

(32)

It follows that by using the hypotheses (H1) (H3) (ie (23)(21) resp) and (H4)

int

119861119877(1205850)

[119860

120572

119894(120585 119906 119883119906)

minus119860

120572

119894(120585 119906 119901

0)] 120601

2

(119883119906 minus 119901

0) 119889120585

= 2int

119861119877(1205850)

[119860

120572

119894(120585 119906 119901

0)

minus119860

120572

119894(120585 119906 119883119906)] 120601V119883120601119889120585

+ int

119861119877(1205850)

[119860

120572

119894(120585 119906

0+ (120585

1

minus 120585

1

0) 119901

0 119901

0)

minus119860

120572

119894(120585 119906 119901

0) ]119883120593

120572

119889120585

+ int

119861119877(1205850)

[119860

120572

119894(120585

0 119906

0 119901

0)

minus119860

120572

119894(120585 119906

0+ (120585

1

minus 120585

1

0) 119901

0 119901

0)]119883120593

120572

119889120585

+ int

119861119877(1205850)

119861

120572

(120585 119906 119883119906) 120593

120572

119889120585

le 119868 + 119868119868 + 119868119868119868 + 119868119881 + 119881

(33)

where

119868 = 2119871int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

120601

1003816

1003816

1003816

1003816

|V| 1003816100381610038161003816

119883120601

1003816

1003816

1003816

1003816

119889120585

119868119868 = (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

) 120581 (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+ 119877

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

times int

119861119877(1205850)

radic120578 (|V|2) 100381610038161003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

120601

2

119889120585

119868119868119868 = 2 (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

) 120581 (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+ 119877

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

times int

119861119877(1205850)

radic120578 (|V|2) |V| 100381610038161003816

1003816

119883120601

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

120601

1003816

1003816

1003816

1003816

119889120585

119868119881 = (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

) 120581 (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+ 119877

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

times int

119861119877(1205850)

radic120578 (119877

2

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

2

)) [

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

120601

2

+2

1003816

1003816

1003816

1003816

120601

1003816

1003816

1003816

1003816

|V| 1003816100381610038161003816

119883120601

1003816

1003816

1003816

1003816

] 119889120585

119881 = 119862int

119861119877(1205850)

(1 + |119906|

119903minus1

+ |119883119906|

2(1minus1119903)

) 120593119889120585

(34)

Applying (H2) the left hand side of (33) can be estimated as

120582int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

120601

2

119889120585

le int

119861119877(1205850)

[119860

120572

119894(120585 119906 119883119906) minus 119860

120572

119894(120585 119906 119901

0)] 120601

2

(119883119906 minus 119901

0) 119889120585

(35)

For 120576 gt 0 to be fixed later we have using Youngrsquos inequality

119868 le 120576int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

21003816

1003816

1003816

1003816

120601

1003816

1003816

1003816

1003816

2

119889120585 +

119862119871

2

(119877 minus 120588)

2int

119861119877(1205850)

|V|2119889120585

(36)

Using Jensenrsquos inequality (20) and the fact that 120578(119905119904) le 119905120578(119904)for 119905 ge 1 we arrive at

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

120581

2

(sdot) int

119861119877(1205850)

120578 (|V|2) 119889120585

le 120596

119899119877

119876minus2

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

120581

2

(sdot) 119877

2

120578 (∮

119861119877(1205850)

|V|2119889120585)

le 120596

119899119877

119876minus2

[∮

119861119877(1205850)

|V|2119889120585

+ (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

120581

2

(sdot) 119877

2

120578

times ((1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

120581

2

(sdot) 119877

2

) ]

le 119877

minus2

int

119861119877(1205850)

|V|2119889120585

+ 120596

119899119877

119876

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

4

120581

4

(sdot) 120578 (119877

2

)

(37)

where 120581(sdot) is an abbreviation of the function 120581(|1199060| + |119901

0|)

Also note that the application of (20) in the second lastinequality is possible by our choice 119877 le 120588

1(|119906

0| + |119901

0|)

6 Abstract and Applied Analysis

Using Youngrsquos inequality and (37) in 119868119868 we obtain

119868119868 le 120576int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

21003816

1003816

1003816

1003816

120601

1003816

1003816

1003816

1003816

2

119889120585

+ 120576

minus1

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

120581

2

(sdot) int

119861119877(1205850)

120578 (|V|2) 119889120585

le 120576int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

21003816

1003816

1003816

1003816

120601

1003816

1003816

1003816

1003816

2

119889120585

+

1

120576(119877 minus 120588)

2int

119861119877(1205850)

|V|2119889120585

+ 120576

minus1

120596

119899119877

119876

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

4

120581

4

(sdot) 120578 (119877

2

)

(38)

And similarly we see

119868119868119868 le

4119862

(119877 minus 120588)

2int

119861119877(1205850)

|V|2119889120585

+ (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

120581

2

(sdot) int

119861119877(1205850)

120578 (|V|2) 119889120585

le

119862

(119877 minus 120588)

2int

119861119877(1205850)

|V|2119889120585

+ 120596

119899119877

119876

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

4

120581

4

(sdot) 120578 (119877

2

)

119868119881 le 120576int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

21003816

1003816

1003816

1003816

120601

1003816

1003816

1003816

1003816

2

119889120585

+

4119862120576

(119877 minus 120588)

2int

119861119877(1205850)

|V|2119889120585

+ 120576

minus1

120596

119899119877

119876

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

120581

2

(sdot) 120578

times (∮

119861119877(1205850)

119877

2

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

2

) 119889120585)

le 120576int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

21003816

1003816

1003816

1003816

120601

1003816

1003816

1003816

1003816

2

119889120585

+

119862120576

(119877 minus 120588)

2int

119861119877(1205850)

|V|2119889120585

+ 120576

minus1

120596

119899119877

119876

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

4

120581

4

(sdot) 120578 (119877

2

)

(39)

Here we have used 120581 ge 1 in the last inequality

By Holderrsquos inequality (19) and Youngrsquos inequality onegets

119881 le 119862(int

119861119877(1205850)

1003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119903

119889120585)

1119903

times (int

119861119877(1205850)

(1 + |119906|

119903

+ |119883119906|

2

) 119889120585)

(119903minus1)119903

le 119862(int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

2

119889120585)

12

times (int

119861119877(1205850)

(1 + |119906|

119903

+ |119883119906|

2

) 119889120585)

(119903minus1)119903

le 120576int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

2

119889120585 + 119862 (120576)

times (int

119861119877(1205850)

(1 + |119906|

119903

+ |119883119906|

2

) 119889120585)

2(119903minus1)119903

le 120576int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

21003816

1003816

1003816

1003816

120601

1003816

1003816

1003816

1003816

2

119889120585

+

119862120576

(119877 minus 120588)

2int

119861119877(1205850)

|V|2119889120585

+ 119862 (120576) (int

119861119877(1205850)

(1 + |119906|

119903

+ |119883119906|

2

) 119889120585)

2(119903minus1)119903

(40)

where we have used the fact that |119883120593| le 120601|119883119906 minus 1199010| + 119862(119877 minus

120588)|V|Applying these estimates to (37) we obtain

(120582 minus 4120576) int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

120601

2

119889120585

le

119862 (119871 120576)

(119877 minus 120588)

2int

119861119877(1205850)

|V|2119889120585

+ (120576

minus1

+ 2)120596

119899119877

119876

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

4

120581

4

(sdot) 120578 (119877

2

)

+ 119862 (120576) (int

119861119877(1205850)

(1 + |119906|

119903

+ |119883119906|

2

) 119889120585)

2(119903minus1)119903

(41)

Choosing 120576 = 1205828 we obtain the desired inequality (29)

4 Proof of the Main Theorem

In this section we will complete the proof of the partialregularity results via the following lemmas In the sequel wealways suppose that 119906 isin 11986711988212(ΩR119873) is a weak solution to(1) with the assumptions of (H1)ndash(H4) and (1205831)ndash(1205833)

Abstract and Applied Analysis 7

Lemma 4 Let 119861120588(120585

0) subsub Ω with 120588 le 120588

1(|119906

0| |119901

0|) and 120593 isin

119862

infin

0(119861

120588(120585

0)R119873) satisfying |120593| le 120588

2 and sup119861120588(1205850)

|119883120593| le 1Then there exists a constant 119862

1ge 1 such that

∮

119861120588(1205850)

119860

120572

119894119901119895

120573

(120585

0 119906

0 119901

0) (119883119906 minus 119901

0)119883120593

120572

119889120585

le 119862

1[Φ (120585

0 120588 119901

0)

+ 120596 (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

Φ (120585

0 120588 119901

0))Φ

12

(120585

0 120588 119901

0)

+ 119870

1(

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)radic120578 (120588

2

)] sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

(42)

Proof Using the fact that int119861120588(1205850)

119860

120572

119894(120585

0 119906

0 119901

0)119883120593

120572

119889120585 = 0 andthe weak form (8) we deduce

∮

119861120588(1205850)

[int

1

0

119860

120572

119894119901119895

120573

(120585

0 119906

0 120579119883119906 + (1 minus 120579) 119901

0)

times (119883119906 minus 119901

0) 119889120579]119883120593

120572

119889120585

= ∮

119861120588(1205850)

[119860

120572

119894(120585

0 119906

0 119883119906)

minus119860

120572

119894(120585

0 119906

0 119901

0)]119883120593

120572

119889120585

= ∮

119861120588(1205850)

[119860

120572

119894(120585

0 119906

0 119883119906)

minus119860

120572

119894(120585 119906 119883119906)]119883120593

120572

119889120585

+ ∮

119861120588(1205850)

119861

120572

(120585 119906 119883119906) 120593

120572

119889120585

(43)

It yields

∮

119861120588(1205850)

119860

120572

119894119901119895

120573

(120585

0 119906

0 119901

0) (119883119906 minus 119901

0)119883120593

120572

119889120585

= ∮

119861120588(1205850)

[int

1

0

(119860

120572

119894119901119895

120573

(120585

0 119906

0 119901

0)

minus119860

120572

119894119901119895

120573

(120585

0 119906

0 120579119883119906 + (1 minus 120579) 119901

0))

times (119883119906 minus 119901

0) 119889120579] 119889120585 sup

119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

+ ∮

119861120588(1205850)

[119860

120572

119894(120585

0 119906

0 119883119906)

minus119860

120572

119894(120585 119906

0+ 119901

0(120585 minus 120585

0) 119883119906)] sup

119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

+ ∮

119861120588(1205850)

[119860

120572

119894(120585 119906

0+ 119901

0(120585 minus 120585

0) 119883119906)

minus119860

120572

119894(120585 119906 119883119906)] sup

119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

+ ∮

119861120588(1205850)

119861

120572

(120585 119906 119883119906) 120593

120572

119889120585

= 119868

1015840

+ 119868119868

1015840

+ 119868119868119868

1015840

+ 119868119881

1015840

(44)

Using (22) Holderrsquos inequality the fact that 119905 rarr 120596

2

(119904 119905) isconcave and Jensenrsquos inequality we have

119868

1015840

le sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

∮

119861120588(1205850)

120596 (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

119889120585

le sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

[∮

119861120588(1205850)

120596

2

(

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

) 119889120585]

12

times[∮

119861120588(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

119889120585]

12

le 120596(

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

Φ (120585

0 120588 119901

0))Φ

12

(120585

0 120588 119901

0) sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

(45)

Similarly using (21) and the fact that 120578(119905119904) le 119905120578(119904) for 119905 ge 1we obtain

119868119868

1015840

le sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

120581 (sdot)radic120578 (120588

2

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

2

))

times ∮

119861120588(1205850)

(1 + |119883119906|) 119889120585

le sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

120581 (sdot)radic120578 (120588

2

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

)

times ∮

119861120588(1205850)

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

) 119889120585

le sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

[(∮

119861120588(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

119889120585)

+ 120581

2

(sdot) (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

120578 (120588

2

)

+ 120581 (sdot) (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

3

radic120578 (120588

2

) ]

le [Φ (120585

0 120588 119901

0)+ 2120581

2

(sdot) (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

3

radic120578 (120588

2

)]

times sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

(46)

8 Abstract and Applied Analysis

where we have used the fact that 120578(1205882) le

radic

120578(120588

2

) whichfollows from the nondecreasing property of the function 120578(119905)(1205784) and our assumption 120588 le 120588

1le 1

In the same way it follows that by using (21) (37) and(19)

119868119868119868

1015840

le sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

120581 (sdot)∮

119861120588(1205850)

radic120578 (|V|2) (1 + |119883119906|) 119889120585

le sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

[∮

119861120588(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

119889120585

+ 120581

2

(sdot)∮

119861120588(1205850)

120578 (|V|2) 119889120585

+ 120581 (sdot) (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)∮

119861120588(1205850)

radic120578 (|V|2) 119889120585]

le sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

[Φ (120585

0 120588 119901

0)

+ 2120588

minus2

∮

119861120588(1205850)

|V|2119889120585 + 1205814 (sdot) 120578 (1205882)

+ 120581

2

(sdot) (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

radic120578 (120588

2

) ]

le [(1 + 2119862

119901)Φ (120585

0 120588 119901

0)

+ 2120581

4

(sdot) (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

radic120578 (120588

2

)]

(47)

Using Holderrsquos inequality (19) and Youngrsquos inequality wehave

119868119881

1015840

le 119862∮

119861120588(1205850)

(1 + |119906|

119903minus1

+ |119883119906|

2(1minus1119903)

)

1003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119889120585

le 119862∮

119861120588(1205850)

|119883119906|

2(1minus1119903) 10038161003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119889120585

+ 119862∮

119861120588(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus 119906

0minus 119901

0(120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

119903minus11003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119889120585

+ 119862120588

2

[1 + (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

119903minus1

]

le 119862(∮

119861120588(1205850)

|119883119906|

2

119889120585)

(1minus1119903)

(∮

119861120588(1205850)

1003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119903

119889120585)

1119903

+ 119862(∮

119861120588(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus 119906

0minus 119901

0(120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

119903

119889120585)

(1minus1119903)

times (∮

119861120588(1205850)

1003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119903

119889120585)

1119903

+ 119862120588

2

[1 + (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

119903minus1

]

le 119862(∮

119861120588(1205850)

|119883119906|

2

119889120585)

(1minus1119903)

(∮

119861120588(1205850)

1003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119903

119889120585)

1119903

+ (∮

119861120588(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

119889120585)

(1199032)(1minus1119903)

times (∮

119861120588(1205850)

1003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119903

119889120585)

1119903

+ 119862120588

2

[1 + (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

119903minus1

]

le 119862119906

11986711988212(∮

119861120588(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

119889120585)

(1minus1119903)

times (∮

119861120588(1205850)

1003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119903

119889120585)

1119903

+ 119862120588 [1 + (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

119903minus1

]

le 119862119906

11986711988212(119861120588(1205850))

(∮

119861120588(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

119889120585)

+ 119862120588

2119903

+ 119862120588

2

[1 + (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

119903minus1

]

le 119862

2Φ(120585

0 120588 119901

0) + 119862120588

2

(1 +

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

119903minus1

le 119862

2Φ(120585

0 120588 119901

0) + 119862120581 (sdot) (1 +

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

radic120578 (120588

2

)

(48)

where we have used the assumption (1205784) and the fact that 119903 =2119876(119876 minus 2) = (2119899 + 4)2119899 le 3 and 119862

2= 119862119906

11986711988212(119861120588(1205850))

ge

1 Combining these estimates we obtain the conclusion with119862

1= (1 + 119862

2+ 2119862

119901) ge 1

Lemma 5 Assume that the conditions of Lemma 2 and thefollowing smallness conditions hold

120596 (

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

1003816

(119883119906)

1205850120588

1003816

1003816

1003816

1003816

1003816

Φ (120585

0 120588 (119883119906)

1205850120588))

+ Φ

12

(120585

0 120588 (119883119906)

1205850120588) le

120575

2

(49)

119862

3119870

2

1(

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

1205850120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

) le 120575

2 (50)

with 1198623= 8119862

2

1119862

5 together with

120588 le 120588

1(1 +

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

1 +

1003816

1003816

1003816

1003816

1003816

(119883119906)

1205850120588

1003816

1003816

1003816

1003816

1003816

) (51)

Then the following growth condition holds for 120591 isin [120574 1)

Φ (120585

0 120579120588) le 120579

2120591

Φ(120585

0 120588)

+ 119870

lowast

(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

)

(52)

where one abbreviates Φ(1205850 119903) = Φ(120585

0 119903 (119883119906)

1205850119903) and

119870

lowast

(119904 119905) = 119870(119904 119905) + (2 + 119904 + 119905)

2(119903minus1) with 119870(119904 119905) = (4120575

minus2

+

2

119876

119862

119888)119870

2

1(1 + 119904 1 + 119905)

Proof We define 119908 = [119906 minus 1199061205850120588

minus 119901

0(120585

1

minus 120585

1

0)]120590

minus1 where

120590 = 119862

1radic

Φ(120585

0 120588 119901

0) + 4120575

minus2119870

2

1(

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

) 120578 (120588

2)

(53)

Abstract and Applied Analysis 9

Then we have 119883119908 = 120590

minus1

(119883119906 minus 119901

0) Now we consider

119861

120588(120585

0) subsub Ω such that 120588 le 120588

1(|119906

0| |119901

0|) Applying Lemma 4

on 119861120588(120585

0) to 119906 we have for any 120593 isin 119862infin

0(119861

120588(120585

0)R119873)

∮

119861120588(1205850)

|119883119908|

2

119889120585 = 120590

minus2

Φ(120585

0 120588 119901

0) le

1

119862

2

1

le 1 (54)

∮

119861120588(1205850)

119860

120572

119894119901119895

120573

(120585

0 119906

0 119901

0)119883119908119883120593119889120585

le [Φ

12

(120585

0 120588 119901

0)

+120596 (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

Φ (120585

0 120588 119901

0)) +

120575

2

] sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

(55)

In consideration of the small condition (49) we see that(54) and (55) imply conditions (26) in Lemma 2 Also notethat (H1) and (H3) imply condition (25) So there exists an119860

120572

119894119901119895

120573

(120585

0 119906

0 119901

0)-harmonic function ℎ isin 11986711988212(119861

120588(120585

0)R119873)

such that

∮

119861120588(1205850)

|119883ℎ|

2

119889120585 le 1 120588

minus2

∮

119861120588(1205850)

|119908 minus ℎ|

2

119889120585 le 120576 (56)

Taking 1199060= 119906

12058502120579120588 120579 isin (0 14] and replacing 119901

0by 1199010+

120590(119883ℎ)

1205850 2120579120588 we use Lemma 3 to obtain

int

119861120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119883119906 minus 119901

0minus 120590(119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

2

119889120585

le 119862

119888[

1

(120579120588)

2int

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus 119906

12058502120579120588

minus (119901

0+ 120590(119883ℎ)

12058502120579120588)

times (120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

2

119889120585 + 119865]

(57)

where

119865 = 120596

119899(2120579120588)

119876

119870

1(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119901

0+ 120590(119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 ((2120579120588)

2

)

+ [int

1198612120579120588(1205850)

(1 + 119906

119903

+ |119883119906|

2

) 119889120585]

2(1minus1119903)

(58)

Using the fact that 119906 minus (1199010+ 120590(119883ℎ)

12058502120579120588)(120585

1

minus 120585

1

0) has mean

value 11990612058502120579120588

on the ball 1198612120579120588(120585

0) the definition of119908 and (19)

we have

1

(120579120588)

2∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus 119906

1205850 2120579120588minus (119901

0+ 120590(119883ℎ)

12058502120579120588) (120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

2

119889120585

le

4120590

2

(2120579120588)

2∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119908 minus ℎ

1205850 2120579120588

minus(119883ℎ)

12058502120579120588(120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

2

119889120585

le

4120590

2

(2120579120588)

2[∮

1198612120579120588(1205850)

|119908 minus ℎ|

2

119889120585

+ ∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

ℎ minus ℎ

1205850 2120579120588

minus(119883ℎ)

12058502120579120588(120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

2

119889120585]

le 4120590

2

[(2120579)

minus119876minus2

120576 + 119862

119901∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119883ℎ minus (119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

2

119889120585]

le 4120590

2

[(2120579)

minus119876minus2

120576 + 119862

2

119901(2120579120588)

2

∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119883

2

ℎ

1003816

1003816

1003816

1003816

1003816

2

119889120585]

le 4120590

2

[(2120579)

minus119876minus2

120576 + 119862

2

119901(2120579)

2

119862

0]

le 119862

4(120579

minus119876minus2

120576 + 120579

2

) [Φ (120585

0 120588 119901

0)

+ 4120575

minus2

119870

2

1(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

) 120578 (120588

2

)]

(59)

where 1198624= 119862

4(119876 120582 119871) ge 1 Note that in the second last

inequality we have used the fact that

∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119883

2

ℎ

1003816

1003816

1003816

1003816

1003816

119889120585 le sup119861120588(1205850)

1003816

1003816

1003816

1003816

1003816

119883

2

ℎ

1003816

1003816

1003816

1003816

1003816

le 119862

0120588

minus2

∮

119861120588(1205850)

|119883ℎ|

2

119889120585 le 119862

0120588

minus2

(60)

In consideration of the fact that 119903 = 2119876(119876 minus 2) gt 2 119876 ge 4

and the assumptions 120579 isin (0 14] and Φ le 1 it follows that

[∮

1198612120579120588(1205850)

(1 + 119906

119903

+ |119883119906|

2

) 119889120585]

2(1minus1119903)

le 119862[∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

119889120585]

2(1minus1119903)

+ 119862(∮

1198612120579120588(1205850)

|119883119906|

2

119889120585)

119903minus1

+ (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

4(1minus1119903)

)

10 Abstract and Applied Analysis

le 119862 [(2120579)

minus2119876(1minus1119903)

Φ(120585

0 120588 119901

0)

2(1minus1119903)

+(2120579)

minus119876(119903minus1)

Φ(120585

0 120588 119901

0)

119903minus1

]

+ (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

2(1minus1119903)

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

4(1minus1119903)

)

le 119862(2120579)

minus119876(119903minus1)

Φ(120585

0 120588 119901

0)

2(1minus1119903)

+ (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

2(1minus1119903)

)

2

(61)

Let 119875 = 1199010+120590(119883ℎ)

12058502120579120588with 119901

0= (119883119906)

12058502120579120588 Combining

these estimates (57)ndash(61) and considering the small condition(51) (it implies 120588 le 120588

1(|119906

12058502120579120588| |119875|) see (64) and (65)) we

deduce that

Φ(120585

0 120579120588) le

1003816

1003816

1003816

1003816

1003816

119861

120579120588(120585

0)

1003816

1003816

1003816

1003816

1003816

minus1

H119899int

1198612120579120588(1205850)

|119883119906 minus 119875|

2

119889120585

le 119862

119888

2

119876

(120579120588)

2∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus 119906

12058502120579120588

minus (119901

0+ 120590(119883ℎ)

12058502120579120588)

times (120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

2

119889120585

+ 2

119876

119862

119888119870

1(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119901

0+ 120590(119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

)

+ 119862

119888

(2120579120588)

2119876(1minus1119903)

(120579120588)

119876

times[∮

1198612120579120588(1205850)

(1 + 119906

119903

+ |119883119906|

2

) 119889120585]

2(1minus1119903)

le 2

119876

119862

4119862

119888(120579

minus119876minus2

120576 + 120579

2

)

times [Φ (120585

0 120588) + 4120575

minus2

119870

2

1

times (

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

) ]

+ 2

119876

119862

119888119870

1(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

+120590(119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

)

+ [2

119876

119862

119888(2120579)

2minus119876(119903minus1)

Φ(120585

0 120588)

2(1minus1119903)

+(1 +

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

2(1minus1119903)

)

2

] 120588

2

(62)

Wenow specify 120576 = 120579119876+4 120579 isin (0 14] such that 2119876+11198624119862

119888120579

2

le

120579

2120591 Note that the small condition (50) implies 12059021198625le 1 with

119862

5= max119862

0 119862

1198882

119876

(2120579)

minus(1198762+4)(119876minus2)

and it yields

2

119876

119862

119888(2120579)

2minus119876(119903minus1)

Φ(120585

0 120588)

2(1minus1119903)

le 1

(63)1003816

1003816

1003816

1003816

1003816

120590(119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

le 120590 sup1198612120579120588(1205850)

|119883ℎ|

le 120590radic119862

0(∮

119861120588(1205850)

|119883ℎ|

2

119889120585) le 120590radic119862

0le 1

(64)

where we have used the a priori estimate (28) for the A-harmonic function ℎ Furthermore using (19) and recallingthe definition of 120590 and 119862

1 we have

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

le

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588minus 119906

1205850120588

1003816

1003816

1003816

1003816

1003816

le

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+ (2120579)

minus1198762

times (∮

119861120588(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus (119883119906)

1205850120588(120585

1

minus 120585

1

0) minus 119906

1205850120588

1003816

1003816

1003816

1003816

1003816

2

119889120585)

12

le

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+ (2120579)

minus1198762

120588radic119862

119901Φ

12

(120585

0 120588)

le

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+

120590

radic

119862

119901

119862

1(2120579)

1198762

le

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+ 120590radic119862

5le

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+ 1

(65)

Combining these estimates with (62) we have

Φ(120585

0 120579120588) le 120579

2120591

Φ(120585

0 120588)

+ [4120575

minus2

119870

2

1(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

)

+ 2

119876

119862

119888119870

1(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

+120590(119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

)] 120578 (120588

2

)

+ [1 + (1 +

1003816

1003816

1003816

1003816

1003816

(119883119906)

1205850 2120579120588

1003816

1003816

1003816

1003816

1003816

2(1minus1119903)

)

2

] 120578 (120588

2

)

le 120579

2120591

Φ(120585

0 120588) + 119870 (

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

)

+ (2 +

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

)

2(119903minus1)

120578 (120588

2

)

le 120579

2120591

Φ(120585

0 120588) + 119870

lowast

(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

)

(66)

Then the proof of Lemma 5 is complete

Abstract and Applied Analysis 11

For 119879 gt 0 we find Φ0(119879) gt 0 (depending on 119876119873 120582 119871

120591 and 120596) such that

120596

2

(2119879 2Φ

0(119879)) + 2Φ

0(119879) le

1

2

120575

2

119862

1Φ

0(119879) le 120579

119876

(1 minus 120579

120591

)

2

119879

2

(67)

With Φ0(119879) from (67) we choose 120588

0(119879) isin (0 1] (depending

on 119876119873 120582 119871 120591 120596 120578 and 120581) such that

120588

0(119879) le 120588

1(1 + 2119879 1 + 2119879)

119862

3119870

2

1(2119879 2119879) 120578 (120588

0(119879)

2

) le 120575

2

119870

0(119879) 120578 (120588

0(119879)

2

) le (120579

2120574

minus 120579

2120591

)Φ

0(119879)

2 (1 + 119862

119901)119870

0(119879)119867 (120588

0(119879)

2

) le 120579

119876

(1 minus 120579

120574

)

2

(120579

2120574

minus 120579

2120591

) 119879

2

(68)

where1198700(119879) = 119870

lowast

(2119879 2119879)By the proof method of of Lemma 51 in [12] and

conditions (67)-(68) Lemma 6 can be proved As is wellknown it is sufficient to complete the proof ofTheorem 1 oncewe obtain Lemma 6

Lemma 6 Assume that for some 1198790gt 0 and 119861

120588(120585

0) subsub Ω one

has

(1) |1199061205850120588| + |(119883119906)

1205850120588| le 119879

0

(2) 120588 le 1205880(119879

0)

(3) Φ(1205850 120588) le Φ

0(119879

0)

Then the small conditions (49)ndash(51) are satisfied on the balls119861

120579119895120588(120585

0) for 119895 isin 119873 cup 0 Moreover the limit Λ

1205850=

lim119895rarrinfin

(119883119906)

1205850120579119895120588exists and the inequality

∮

119861120588(1205850)

1003816

1003816

1003816

1003816

1003816

119883119906 minus Λ

1205850

1003816

1003816

1003816

1003816

1003816

2

119889120585 le 119862

6((

119903

120588

)

2120591

Φ(120585

0 120588) + 119867 (119903

2

))

(69)

is valid for 0 lt 119903 le 120588 with a constant 1198626= 119862

6(119876 119873 120582 119871

120591 119886119899119889 119879

0)

Proof The proof is very similar to the proof of Lemma 51 in[12] We omit it here

Acknowledgments

The project was supported by the National Natural ScienceFoundation of China (no 11201081 and no 11126294) andby the Science and Technology Planning Project of JiangxiProvince China no GJJ13657

References

[1] J Wang and P Niu ldquoOptimal partial regularity for weaksolutions of nonlinear sub-elliptic systems in Carnot groupsrdquoNonlinear Analysis Theory Methods amp Applications vol 72 no11 pp 4162ndash4187 2010

[2] J Wang and D Liao ldquoOptimal partial regularity for sub-elliptic systems with sub-quadratic growth in Carnot groupsrdquoNonlinear Analysis Theory Methods amp Applications vol 75 no4 pp 2499ndash2519 2012

[3] E de Giorgi ldquoUn esempio di estremali discontinue per unproblema variazionale di tipo ellitticordquo Bollettino della UnioneMatematica Italiana vol 4 pp 135ndash137 1968

[4] M Giaquinta Multiple Integrals in the Calculus of Variationsand Nonlinear Elliptic Systems Princeton University PressPrinceton NJ USA 1983

[5] Y Chen and L Wu Second Order Elliptic Equations and EllipticSystems Science Press Beijing China 2003

[6] F Duzaar and K Steffen ldquoOptimal interior and boundary reg-ularity for almost minimizers to elliptic variational integralsrdquoJournal fur die Reine und Angewandte Mathematik vol 546 pp73ndash138 2002

[7] F Duzaar and J F Grotowski ldquoOptimal interior partial regu-larity for nonlinear elliptic systems the method of A-harmonicapproximationrdquo Manuscripta Mathematica vol 103 no 3 pp267ndash298 2000

[8] F Duzaar J F Grotowski and M Kronz ldquoRegularity ofalmost minimizers of quasi-convex variational integrals withsubquadratic growthrdquo Annali di Matematica Pura ed ApplicataIV vol 184 no 4 pp 421ndash448 2005

[9] F Duzaar and G Mingione ldquoThe p-harmonic approximationand the regularity of p-harmonic mapsrdquo Calculus of Variationsand Partial Differential Equations vol 20 no 3 pp 235ndash2562004

[10] F Duzaar and G Mingione ldquoRegularity for degenerate ellipticproblems via p-harmonic approximationrdquo Annales de lrsquoInstitutHenri Poincare Analyse Non Lineaire vol 21 no 5 pp 735ndash7662004

[11] S Chen andZ Tan ldquoThemethod ofA-harmonic approximationand optimal interior partial regularity for nonlinear ellipticsystems under the controllable growth conditionrdquo Journal ofMathematical Analysis and Applications vol 335 no 1 pp 20ndash42 2007

[12] F Duzaar and A Gastel ldquoNonlinear elliptic systems with Dinicontinuous coefficientsrdquo Archiv der Mathematik vol 78 no 1pp 58ndash73 2002

[13] F Duzaar A Gastel and G Mingione ldquoElliptic systemssingular sets and Dini continuityrdquo Communications in PartialDifferential Equations vol 29 no 7-8 pp 1215ndash1240 2004

[14] Y Qiu and Z Tan ldquoOptimal interior partial regularity fornonlinear elliptic systems with Dini continuous coefficientsrdquoActa Mathematica Scientia B vol 30 no 5 pp 1541ndash1554 2010

[15] Y Qiu ldquoOptimal partial regularity of second order nonlin-ear elliptic systems with Dini continuous coefficients for thesuperquadratic caserdquo Nonlinear Analysis Theory Methods ampApplications vol 75 no 8 pp 3574ndash3590 2012

[16] L Capogna ldquoRegularity of quasi-linear equations in theHeisen-berg grouprdquoCommunications on Pure andAppliedMathematicsvol 50 no 9 pp 867ndash889 1997

[17] L Capogna ldquoRegularity for quasilinear equations and 1-quasiconformal maps in Carnot groupsrdquo MathematischeAnnalen vol 313 no 2 pp 263ndash295 1999

[18] S Marchi ldquo1198621120572 local regularity for the solutions of the p-Laplacian on the Heisenberg group for 2 lt 119901 lt 1 +

radic

5rdquoZeitschrift fur Analysis und ihre Anwendungen vol 20 no 3 pp617ndash636 2001

12 Abstract and Applied Analysis

[19] S Marchi ldquo1198621120572 local regularity for the solutions of the p-Laplacian on the Heisenberg groupThe case 1+1radic5 lt 119901 le 2rdquoCommentationes Mathematicae Universitatis Carolinae vol 44no 1 pp 33ndash56 2003

[20] S Marchi ldquoL119901 regularity of the derivative in the secondcommutator direction for nonlinear elliptic equations on theHeisenberg grouprdquo Accademia Nazionale delle Scienze detta deiXL Rendiconti Serie V Memorie di Matematica e ApplicazioniParte I vol 26 pp 1ndash15 2002

[21] A Domokos ldquoDifferentiability of solutions for the non-degenerate p-Laplacian in the Heisenberg grouprdquo Journal ofDifferential Equations vol 204 no 2 pp 439ndash470 2004

[22] A Domokos On the regularity of p-harmonic functions inthe Heisenberg group [PhD thesis] University of PittsburghPittsburgh Pa USA 2004

[23] J J Manfredi and G Mingione ldquoRegularity results for quasilin-ear elliptic equations in the Heisenberg grouprdquo MathematischeAnnalen vol 339 no 3 pp 485ndash544 2007

[24] G Mingione A Zatorska-Goldstein and X Zhong ldquoGradientregularity for elliptic equations in the Heisenberg grouprdquoAdvances in Mathematics vol 222 no 1 pp 62ndash129 2009

[25] L Capogna and N Garofalo ldquoRegularity of minimizers of thecalculus of variations inCarnot groups via hypoellipticity of sys-tems ofHormander typerdquo Journal of the EuropeanMathematicalSociety vol 5 no 1 pp 1ndash40 2003

[26] E Shores ldquoHypoellipticity forlinear degenerate elliptic systemsin Carnot groups and applicationsrdquo httparxivorgabsmath0502569

[27] A Foglein ldquoPartial regularity results for subelliptic systemsin the Heisenberg grouprdquo Calculus of Variations and PartialDifferential Equations vol 32 no 1 pp 25ndash51 2008

[28] G Lu ldquoThe sharp Poincare inequality for free vector fields anendpoint resultrdquo Revista Matematica Iberoamericana vol 10no 2 pp 453ndash466 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 4: Research Article Partial Regularity for Nonlinear Subelliptic Systems …downloads.hindawi.com/journals/aaa/2013/950134.pdf · 2019-07-31 · Partial Regularity for Nonlinear Subelliptic

4 Abstract and Applied Analysis

in (11) fit Hormanderrsquos vector fields and that (19) is valid for119901 = 119902 = 2

Following [12] for technical convenience letting 120578(119905) =120583

2

(

radic

2119905) we have the corresponding properties for 120578 (1205781) 120578is continuous nondecreasing and 120578(0) = 0 (1205782) 120578 is concaveand 119903 rarr 119903

minus120574

120578(119903) is nonincreasing for some exponent 120574 isin(0 1) (1205783)119867(119903) = 41198722(radic2119903) = [int119903

0

(

radic

120578(120588)120588)119889120588]

2

lt infin forsome 119903 gt 0 Changing 120581 by a constant but keeping 120581 ge 1 wemay assume the following (1205784) 120578(1) = 1 implying 120578(119905) ge 119905for 119905 isin [0 1] Also note that it implies that from (1205782) and (1205784)120578(119905) le (120574

2

4)119867(119905) for all 119905 ge 0Furthermore the following inequality holds

119904120578 (119905) le 119904120578 (119904) + 119905 119904 isin [0 1] 119905 gt 0 (20)

The condition (H3) becomes

1003816

1003816

1003816

1003816

1003816

119860

120572

119894(120585 119906 119901) minus 119860

120572

119894(

120585 119901)

1003816

1003816

1003816

1003816

1003816

le 120581 (|119906|)radic120578 (119889

2

(120585

120585) + |119906 minus |

2

) (1 +

1003816

1003816

1003816

1003816

119901

1003816

1003816

1003816

1003816

)

(21)

Moreover we deduce the existence of a nonnegative modulusof continuity with 120596(119905 0) = 0 for all 119905 such that 120596(119904 119905) isnondecreasing with respect to 119905 for fixed 119904 and 1205962(119904 119905) isconcave and nondecreasing with respect to 119904 for fixed 119905 Alsowe have for |119906| + |119883119906| le 119872

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119860

120572

119894119901119895

120573

(120585 119906 119901) minus 119860

120572

119894119901119895

120573

(

120585

119901)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

le 120596 (119872 119889

2

(120585

120585) + |119906 minus |

2

+

1003816

1003816

1003816

1003816

119901 minus

119901

1003816

1003816

1003816

1003816

2

)

(22)

Using (H1) and (H2) we see that

1003816

1003816

1003816

1003816

119860

120572

119894(120585 119906 119901) minus 119860

120572

119894(120585 119906

119901)

1003816

1003816

1003816

1003816

le 119871

1003816

1003816

1003816

1003816

119901 minus

119901

1003816

1003816

1003816

1003816

(23)

(119860

120572

119894(120585 119906 119901) minus 119860

120572

119894(120585 119906

119901)) (119901 minus

119901) ge 120582

1003816

1003816

1003816

1003816

119901 minus

119901

1003816

1003816

1003816

1003816

2

(24)

In the sequel the constant 119862may vary from line to line

3 Caccioppoli Type Inequality

In this section we present the followingA-harmonic approx-imation lemma in the Heisenberg group introduced byFoglein [27] with 119901 = 2 as a special case and prove aCaccioppoli type inequality in our setting

Lemma 2 Let 120582 and 119871 be fixed positive numbers and 119899119873 isin

N with 119899 ge 2 If for any given 120576 gt 0 there exists 120575 =

120575(119899119873 120582 120576) isin (0 1] with the following properties

(I) for anyA isin Bil(R2119899119873) satisfying

A (] ]) ge 120582|]|2 A (] ]) le 119871 |]| |]| ] ] isin R2119899119873

(25)

(II) for any 119908 isin 11986711988212(119861120588(120585

0)R119873) satisfying

∮

119861120588(1205850)

|119883119908|

2

119889120585 le 1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

∮

119861120588(1205850)

A (119883119908119883120593) 119889120585

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

le 120575 sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

forall120593 isin 119862

1

0(119861

120588(120585

0) R119873

)

(26)

then there exists anA-harmonic function ℎ such that

∮

119861120588(1205850)

|119883ℎ|

2

119889120585 le 1 120588

minus2

∮

119861120588(1205850)

|ℎ minus 119908|

2

119889120585 le 120576 (27)

Foglein [27] established a priori estimate for the weaksolution 119906 to homogeneous sub-elliptic systemswith constantcoefficients in the Heisenberg group (also see [25] for Carnotgroups of step 2) We list it as follows

sup1198611205882(1205850)

(|119906|

2

+ 120588

2

|119883119906|

2

+ 120588

41003816

1003816

1003816

1003816

1003816

119883

2

119906

1003816

1003816

1003816

1003816

1003816

2

) le 119862

0∮

119861120588(1205850)

|119883119906|

2

119889120585

(28)

In what follows we let 1205881(119904 119905) = (1 + 119904 + 119905)

minus1

120581(119904 + 119905)

minus1 and119870

1(119904 119905) = (1 + 119905)

4

120581(119904 + 119905)

4 for 119904 119905 ge 0 Note that 1205881le 1 and

that 119904 rarr 120588

1(119904 119905) 119905 rarr 120588

1(119904 119905) are nonincreasing functions

Lemma 3 Let 119906 isin 11986711988212(ΩR119873) be a weak solution to thesystem (1) under the conditions (H1)ndash(H4) (1205831)ndash(1205833) Thenfor every 120585

0= (119909

0

1 119909

0

2 119909

0

119899 119910

0

1 119910

0

2 119910

0

119899 119905) isin Ω 119906

0isin R119873

119901

0isin R2119899119873 and 0 lt 120588 lt 119877 lt 120588

1(|119906

0| |119901

0|) le 1 such that

119861

119877(120585

0) subsub Ω the inequality

int

119861120588(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

119889120585

le 119862

119888[

1

(119877 minus 120588)

2int

119861119877(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus 119906

0minus (120585

1

minus 120585

1

0) 119901

0

1003816

1003816

1003816

1003816

1003816

2

119889120585 + 119865]

(29)

holds where 1205851 = (1199091 119909

2 119909

119899 119910

1 119910

2 119910

119899) is the horizon-

tal component of 120585 = (1205851 119905) isin Ω and

119865 = 120596

119899119877

119876

119870

1(

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

) 120578 (119877

2

)

+ [int

119861119877(1205850)

(1 + 119906

119903

+ |119883119906|

2

) 119889120585]

2(1minus1119903)

(30)

Proof Let V = 119906minus1199060minus(120585

1

minus120585

1

0)119901

0 Take a test function120593 = 1206012V

in (8) with 120601 isin 119862infin0(119861

119877(120585

0)R119873) satisfying 0 le 120601 le 1 |nabla120601| le

119862(119877minus120588) and 120601 equiv 1 on 119861120588(120585

0) Then we have119883V = 119883119906minus119901

0

|119883120593| le 120601|119883119906 minus 119901

0| + 119862(119877 minus 120588)|V| and

int

119861119877(1205850)

119860

120572

119894(120585 119906 119883119906) 120601

2

(119883119906 minus 119901

0) 119889120585

= minus2int

119861119877(1205850)

120601119883120601119860

120572

119894(120585 119906 119883119906) V119889120585

+ int

119861119877(1205850)

119861

120572

(120585 119906 119883119906) 120593

120572

119889120585

(31)

Abstract and Applied Analysis 5

Adding this to the equations

minus int

119861119877(1205850)

119860

120572

119894(120585 119906 119901

0) 120601

2

(119883119906 minus 119901

0) 119889120585

= 2int

119861119877(1205850)

120601119883120601119860

120572

119894(120585 119906 119901

0) V119889120585

minus int

119861119877(1205850)

119860

120572

119894(120585 119906 119901

0)119883120593

120572

119889120585

0 = int

119861119877(1205850)

119860

120572

119894(120585

0 119906

0 119901

0)119883120593

120572

(32)

It follows that by using the hypotheses (H1) (H3) (ie (23)(21) resp) and (H4)

int

119861119877(1205850)

[119860

120572

119894(120585 119906 119883119906)

minus119860

120572

119894(120585 119906 119901

0)] 120601

2

(119883119906 minus 119901

0) 119889120585

= 2int

119861119877(1205850)

[119860

120572

119894(120585 119906 119901

0)

minus119860

120572

119894(120585 119906 119883119906)] 120601V119883120601119889120585

+ int

119861119877(1205850)

[119860

120572

119894(120585 119906

0+ (120585

1

minus 120585

1

0) 119901

0 119901

0)

minus119860

120572

119894(120585 119906 119901

0) ]119883120593

120572

119889120585

+ int

119861119877(1205850)

[119860

120572

119894(120585

0 119906

0 119901

0)

minus119860

120572

119894(120585 119906

0+ (120585

1

minus 120585

1

0) 119901

0 119901

0)]119883120593

120572

119889120585

+ int

119861119877(1205850)

119861

120572

(120585 119906 119883119906) 120593

120572

119889120585

le 119868 + 119868119868 + 119868119868119868 + 119868119881 + 119881

(33)

where

119868 = 2119871int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

120601

1003816

1003816

1003816

1003816

|V| 1003816100381610038161003816

119883120601

1003816

1003816

1003816

1003816

119889120585

119868119868 = (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

) 120581 (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+ 119877

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

times int

119861119877(1205850)

radic120578 (|V|2) 100381610038161003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

120601

2

119889120585

119868119868119868 = 2 (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

) 120581 (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+ 119877

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

times int

119861119877(1205850)

radic120578 (|V|2) |V| 100381610038161003816

1003816

119883120601

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

120601

1003816

1003816

1003816

1003816

119889120585

119868119881 = (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

) 120581 (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+ 119877

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

times int

119861119877(1205850)

radic120578 (119877

2

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

2

)) [

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

120601

2

+2

1003816

1003816

1003816

1003816

120601

1003816

1003816

1003816

1003816

|V| 1003816100381610038161003816

119883120601

1003816

1003816

1003816

1003816

] 119889120585

119881 = 119862int

119861119877(1205850)

(1 + |119906|

119903minus1

+ |119883119906|

2(1minus1119903)

) 120593119889120585

(34)

Applying (H2) the left hand side of (33) can be estimated as

120582int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

120601

2

119889120585

le int

119861119877(1205850)

[119860

120572

119894(120585 119906 119883119906) minus 119860

120572

119894(120585 119906 119901

0)] 120601

2

(119883119906 minus 119901

0) 119889120585

(35)

For 120576 gt 0 to be fixed later we have using Youngrsquos inequality

119868 le 120576int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

21003816

1003816

1003816

1003816

120601

1003816

1003816

1003816

1003816

2

119889120585 +

119862119871

2

(119877 minus 120588)

2int

119861119877(1205850)

|V|2119889120585

(36)

Using Jensenrsquos inequality (20) and the fact that 120578(119905119904) le 119905120578(119904)for 119905 ge 1 we arrive at

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

120581

2

(sdot) int

119861119877(1205850)

120578 (|V|2) 119889120585

le 120596

119899119877

119876minus2

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

120581

2

(sdot) 119877

2

120578 (∮

119861119877(1205850)

|V|2119889120585)

le 120596

119899119877

119876minus2

[∮

119861119877(1205850)

|V|2119889120585

+ (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

120581

2

(sdot) 119877

2

120578

times ((1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

120581

2

(sdot) 119877

2

) ]

le 119877

minus2

int

119861119877(1205850)

|V|2119889120585

+ 120596

119899119877

119876

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

4

120581

4

(sdot) 120578 (119877

2

)

(37)

where 120581(sdot) is an abbreviation of the function 120581(|1199060| + |119901

0|)

Also note that the application of (20) in the second lastinequality is possible by our choice 119877 le 120588

1(|119906

0| + |119901

0|)

6 Abstract and Applied Analysis

Using Youngrsquos inequality and (37) in 119868119868 we obtain

119868119868 le 120576int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

21003816

1003816

1003816

1003816

120601

1003816

1003816

1003816

1003816

2

119889120585

+ 120576

minus1

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

120581

2

(sdot) int

119861119877(1205850)

120578 (|V|2) 119889120585

le 120576int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

21003816

1003816

1003816

1003816

120601

1003816

1003816

1003816

1003816

2

119889120585

+

1

120576(119877 minus 120588)

2int

119861119877(1205850)

|V|2119889120585

+ 120576

minus1

120596

119899119877

119876

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

4

120581

4

(sdot) 120578 (119877

2

)

(38)

And similarly we see

119868119868119868 le

4119862

(119877 minus 120588)

2int

119861119877(1205850)

|V|2119889120585

+ (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

120581

2

(sdot) int

119861119877(1205850)

120578 (|V|2) 119889120585

le

119862

(119877 minus 120588)

2int

119861119877(1205850)

|V|2119889120585

+ 120596

119899119877

119876

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

4

120581

4

(sdot) 120578 (119877

2

)

119868119881 le 120576int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

21003816

1003816

1003816

1003816

120601

1003816

1003816

1003816

1003816

2

119889120585

+

4119862120576

(119877 minus 120588)

2int

119861119877(1205850)

|V|2119889120585

+ 120576

minus1

120596

119899119877

119876

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

120581

2

(sdot) 120578

times (∮

119861119877(1205850)

119877

2

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

2

) 119889120585)

le 120576int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

21003816

1003816

1003816

1003816

120601

1003816

1003816

1003816

1003816

2

119889120585

+

119862120576

(119877 minus 120588)

2int

119861119877(1205850)

|V|2119889120585

+ 120576

minus1

120596

119899119877

119876

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

4

120581

4

(sdot) 120578 (119877

2

)

(39)

Here we have used 120581 ge 1 in the last inequality

By Holderrsquos inequality (19) and Youngrsquos inequality onegets

119881 le 119862(int

119861119877(1205850)

1003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119903

119889120585)

1119903

times (int

119861119877(1205850)

(1 + |119906|

119903

+ |119883119906|

2

) 119889120585)

(119903minus1)119903

le 119862(int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

2

119889120585)

12

times (int

119861119877(1205850)

(1 + |119906|

119903

+ |119883119906|

2

) 119889120585)

(119903minus1)119903

le 120576int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

2

119889120585 + 119862 (120576)

times (int

119861119877(1205850)

(1 + |119906|

119903

+ |119883119906|

2

) 119889120585)

2(119903minus1)119903

le 120576int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

21003816

1003816

1003816

1003816

120601

1003816

1003816

1003816

1003816

2

119889120585

+

119862120576

(119877 minus 120588)

2int

119861119877(1205850)

|V|2119889120585

+ 119862 (120576) (int

119861119877(1205850)

(1 + |119906|

119903

+ |119883119906|

2

) 119889120585)

2(119903minus1)119903

(40)

where we have used the fact that |119883120593| le 120601|119883119906 minus 1199010| + 119862(119877 minus

120588)|V|Applying these estimates to (37) we obtain

(120582 minus 4120576) int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

120601

2

119889120585

le

119862 (119871 120576)

(119877 minus 120588)

2int

119861119877(1205850)

|V|2119889120585

+ (120576

minus1

+ 2)120596

119899119877

119876

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

4

120581

4

(sdot) 120578 (119877

2

)

+ 119862 (120576) (int

119861119877(1205850)

(1 + |119906|

119903

+ |119883119906|

2

) 119889120585)

2(119903minus1)119903

(41)

Choosing 120576 = 1205828 we obtain the desired inequality (29)

4 Proof of the Main Theorem

In this section we will complete the proof of the partialregularity results via the following lemmas In the sequel wealways suppose that 119906 isin 11986711988212(ΩR119873) is a weak solution to(1) with the assumptions of (H1)ndash(H4) and (1205831)ndash(1205833)

Abstract and Applied Analysis 7

Lemma 4 Let 119861120588(120585

0) subsub Ω with 120588 le 120588

1(|119906

0| |119901

0|) and 120593 isin

119862

infin

0(119861

120588(120585

0)R119873) satisfying |120593| le 120588

2 and sup119861120588(1205850)

|119883120593| le 1Then there exists a constant 119862

1ge 1 such that

∮

119861120588(1205850)

119860

120572

119894119901119895

120573

(120585

0 119906

0 119901

0) (119883119906 minus 119901

0)119883120593

120572

119889120585

le 119862

1[Φ (120585

0 120588 119901

0)

+ 120596 (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

Φ (120585

0 120588 119901

0))Φ

12

(120585

0 120588 119901

0)

+ 119870

1(

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)radic120578 (120588

2

)] sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

(42)

Proof Using the fact that int119861120588(1205850)

119860

120572

119894(120585

0 119906

0 119901

0)119883120593

120572

119889120585 = 0 andthe weak form (8) we deduce

∮

119861120588(1205850)

[int

1

0

119860

120572

119894119901119895

120573

(120585

0 119906

0 120579119883119906 + (1 minus 120579) 119901

0)

times (119883119906 minus 119901

0) 119889120579]119883120593

120572

119889120585

= ∮

119861120588(1205850)

[119860

120572

119894(120585

0 119906

0 119883119906)

minus119860

120572

119894(120585

0 119906

0 119901

0)]119883120593

120572

119889120585

= ∮

119861120588(1205850)

[119860

120572

119894(120585

0 119906

0 119883119906)

minus119860

120572

119894(120585 119906 119883119906)]119883120593

120572

119889120585

+ ∮

119861120588(1205850)

119861

120572

(120585 119906 119883119906) 120593

120572

119889120585

(43)

It yields

∮

119861120588(1205850)

119860

120572

119894119901119895

120573

(120585

0 119906

0 119901

0) (119883119906 minus 119901

0)119883120593

120572

119889120585

= ∮

119861120588(1205850)

[int

1

0

(119860

120572

119894119901119895

120573

(120585

0 119906

0 119901

0)

minus119860

120572

119894119901119895

120573

(120585

0 119906

0 120579119883119906 + (1 minus 120579) 119901

0))

times (119883119906 minus 119901

0) 119889120579] 119889120585 sup

119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

+ ∮

119861120588(1205850)

[119860

120572

119894(120585

0 119906

0 119883119906)

minus119860

120572

119894(120585 119906

0+ 119901

0(120585 minus 120585

0) 119883119906)] sup

119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

+ ∮

119861120588(1205850)

[119860

120572

119894(120585 119906

0+ 119901

0(120585 minus 120585

0) 119883119906)

minus119860

120572

119894(120585 119906 119883119906)] sup

119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

+ ∮

119861120588(1205850)

119861

120572

(120585 119906 119883119906) 120593

120572

119889120585

= 119868

1015840

+ 119868119868

1015840

+ 119868119868119868

1015840

+ 119868119881

1015840

(44)

Using (22) Holderrsquos inequality the fact that 119905 rarr 120596

2

(119904 119905) isconcave and Jensenrsquos inequality we have

119868

1015840

le sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

∮

119861120588(1205850)

120596 (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

119889120585

le sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

[∮

119861120588(1205850)

120596

2

(

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

) 119889120585]

12

times[∮

119861120588(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

119889120585]

12

le 120596(

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

Φ (120585

0 120588 119901

0))Φ

12

(120585

0 120588 119901

0) sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

(45)

Similarly using (21) and the fact that 120578(119905119904) le 119905120578(119904) for 119905 ge 1we obtain

119868119868

1015840

le sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

120581 (sdot)radic120578 (120588

2

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

2

))

times ∮

119861120588(1205850)

(1 + |119883119906|) 119889120585

le sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

120581 (sdot)radic120578 (120588

2

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

)

times ∮

119861120588(1205850)

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

) 119889120585

le sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

[(∮

119861120588(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

119889120585)

+ 120581

2

(sdot) (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

120578 (120588

2

)

+ 120581 (sdot) (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

3

radic120578 (120588

2

) ]

le [Φ (120585

0 120588 119901

0)+ 2120581

2

(sdot) (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

3

radic120578 (120588

2

)]

times sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

(46)

8 Abstract and Applied Analysis

where we have used the fact that 120578(1205882) le

radic

120578(120588

2

) whichfollows from the nondecreasing property of the function 120578(119905)(1205784) and our assumption 120588 le 120588

1le 1

In the same way it follows that by using (21) (37) and(19)

119868119868119868

1015840

le sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

120581 (sdot)∮

119861120588(1205850)

radic120578 (|V|2) (1 + |119883119906|) 119889120585

le sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

[∮

119861120588(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

119889120585

+ 120581

2

(sdot)∮

119861120588(1205850)

120578 (|V|2) 119889120585

+ 120581 (sdot) (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)∮

119861120588(1205850)

radic120578 (|V|2) 119889120585]

le sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

[Φ (120585

0 120588 119901

0)

+ 2120588

minus2

∮

119861120588(1205850)

|V|2119889120585 + 1205814 (sdot) 120578 (1205882)

+ 120581

2

(sdot) (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

radic120578 (120588

2

) ]

le [(1 + 2119862

119901)Φ (120585

0 120588 119901

0)

+ 2120581

4

(sdot) (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

radic120578 (120588

2

)]

(47)

Using Holderrsquos inequality (19) and Youngrsquos inequality wehave

119868119881

1015840

le 119862∮

119861120588(1205850)

(1 + |119906|

119903minus1

+ |119883119906|

2(1minus1119903)

)

1003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119889120585

le 119862∮

119861120588(1205850)

|119883119906|

2(1minus1119903) 10038161003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119889120585

+ 119862∮

119861120588(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus 119906

0minus 119901

0(120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

119903minus11003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119889120585

+ 119862120588

2

[1 + (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

119903minus1

]

le 119862(∮

119861120588(1205850)

|119883119906|

2

119889120585)

(1minus1119903)

(∮

119861120588(1205850)

1003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119903

119889120585)

1119903

+ 119862(∮

119861120588(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus 119906

0minus 119901

0(120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

119903

119889120585)

(1minus1119903)

times (∮

119861120588(1205850)

1003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119903

119889120585)

1119903

+ 119862120588

2

[1 + (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

119903minus1

]

le 119862(∮

119861120588(1205850)

|119883119906|

2

119889120585)

(1minus1119903)

(∮

119861120588(1205850)

1003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119903

119889120585)

1119903

+ (∮

119861120588(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

119889120585)

(1199032)(1minus1119903)

times (∮

119861120588(1205850)

1003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119903

119889120585)

1119903

+ 119862120588

2

[1 + (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

119903minus1

]

le 119862119906

11986711988212(∮

119861120588(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

119889120585)

(1minus1119903)

times (∮

119861120588(1205850)

1003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119903

119889120585)

1119903

+ 119862120588 [1 + (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

119903minus1

]

le 119862119906

11986711988212(119861120588(1205850))

(∮

119861120588(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

119889120585)

+ 119862120588

2119903

+ 119862120588

2

[1 + (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

119903minus1

]

le 119862

2Φ(120585

0 120588 119901

0) + 119862120588

2

(1 +

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

119903minus1

le 119862

2Φ(120585

0 120588 119901

0) + 119862120581 (sdot) (1 +

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

radic120578 (120588

2

)

(48)

where we have used the assumption (1205784) and the fact that 119903 =2119876(119876 minus 2) = (2119899 + 4)2119899 le 3 and 119862

2= 119862119906

11986711988212(119861120588(1205850))

ge

1 Combining these estimates we obtain the conclusion with119862

1= (1 + 119862

2+ 2119862

119901) ge 1

Lemma 5 Assume that the conditions of Lemma 2 and thefollowing smallness conditions hold

120596 (

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

1003816

(119883119906)

1205850120588

1003816

1003816

1003816

1003816

1003816

Φ (120585

0 120588 (119883119906)

1205850120588))

+ Φ

12

(120585

0 120588 (119883119906)

1205850120588) le

120575

2

(49)

119862

3119870

2

1(

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

1205850120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

) le 120575

2 (50)

with 1198623= 8119862

2

1119862

5 together with

120588 le 120588

1(1 +

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

1 +

1003816

1003816

1003816

1003816

1003816

(119883119906)

1205850120588

1003816

1003816

1003816

1003816

1003816

) (51)

Then the following growth condition holds for 120591 isin [120574 1)

Φ (120585

0 120579120588) le 120579

2120591

Φ(120585

0 120588)

+ 119870

lowast

(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

)

(52)

where one abbreviates Φ(1205850 119903) = Φ(120585

0 119903 (119883119906)

1205850119903) and

119870

lowast

(119904 119905) = 119870(119904 119905) + (2 + 119904 + 119905)

2(119903minus1) with 119870(119904 119905) = (4120575

minus2

+

2

119876

119862

119888)119870

2

1(1 + 119904 1 + 119905)

Proof We define 119908 = [119906 minus 1199061205850120588

minus 119901

0(120585

1

minus 120585

1

0)]120590

minus1 where

120590 = 119862

1radic

Φ(120585

0 120588 119901

0) + 4120575

minus2119870

2

1(

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

) 120578 (120588

2)

(53)

Abstract and Applied Analysis 9

Then we have 119883119908 = 120590

minus1

(119883119906 minus 119901

0) Now we consider

119861

120588(120585

0) subsub Ω such that 120588 le 120588

1(|119906

0| |119901

0|) Applying Lemma 4

on 119861120588(120585

0) to 119906 we have for any 120593 isin 119862infin

0(119861

120588(120585

0)R119873)

∮

119861120588(1205850)

|119883119908|

2

119889120585 = 120590

minus2

Φ(120585

0 120588 119901

0) le

1

119862

2

1

le 1 (54)

∮

119861120588(1205850)

119860

120572

119894119901119895

120573

(120585

0 119906

0 119901

0)119883119908119883120593119889120585

le [Φ

12

(120585

0 120588 119901

0)

+120596 (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

Φ (120585

0 120588 119901

0)) +

120575

2

] sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

(55)

In consideration of the small condition (49) we see that(54) and (55) imply conditions (26) in Lemma 2 Also notethat (H1) and (H3) imply condition (25) So there exists an119860

120572

119894119901119895

120573

(120585

0 119906

0 119901

0)-harmonic function ℎ isin 11986711988212(119861

120588(120585

0)R119873)

such that

∮

119861120588(1205850)

|119883ℎ|

2

119889120585 le 1 120588

minus2

∮

119861120588(1205850)

|119908 minus ℎ|

2

119889120585 le 120576 (56)

Taking 1199060= 119906

12058502120579120588 120579 isin (0 14] and replacing 119901

0by 1199010+

120590(119883ℎ)

1205850 2120579120588 we use Lemma 3 to obtain

int

119861120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119883119906 minus 119901

0minus 120590(119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

2

119889120585

le 119862

119888[

1

(120579120588)

2int

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus 119906

12058502120579120588

minus (119901

0+ 120590(119883ℎ)

12058502120579120588)

times (120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

2

119889120585 + 119865]

(57)

where

119865 = 120596

119899(2120579120588)

119876

119870

1(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119901

0+ 120590(119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 ((2120579120588)

2

)

+ [int

1198612120579120588(1205850)

(1 + 119906

119903

+ |119883119906|

2

) 119889120585]

2(1minus1119903)

(58)

Using the fact that 119906 minus (1199010+ 120590(119883ℎ)

12058502120579120588)(120585

1

minus 120585

1

0) has mean

value 11990612058502120579120588

on the ball 1198612120579120588(120585

0) the definition of119908 and (19)

we have

1

(120579120588)

2∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus 119906

1205850 2120579120588minus (119901

0+ 120590(119883ℎ)

12058502120579120588) (120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

2

119889120585

le

4120590

2

(2120579120588)

2∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119908 minus ℎ

1205850 2120579120588

minus(119883ℎ)

12058502120579120588(120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

2

119889120585

le

4120590

2

(2120579120588)

2[∮

1198612120579120588(1205850)

|119908 minus ℎ|

2

119889120585

+ ∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

ℎ minus ℎ

1205850 2120579120588

minus(119883ℎ)

12058502120579120588(120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

2

119889120585]

le 4120590

2

[(2120579)

minus119876minus2

120576 + 119862

119901∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119883ℎ minus (119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

2

119889120585]

le 4120590

2

[(2120579)

minus119876minus2

120576 + 119862

2

119901(2120579120588)

2

∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119883

2

ℎ

1003816

1003816

1003816

1003816

1003816

2

119889120585]

le 4120590

2

[(2120579)

minus119876minus2

120576 + 119862

2

119901(2120579)

2

119862

0]

le 119862

4(120579

minus119876minus2

120576 + 120579

2

) [Φ (120585

0 120588 119901

0)

+ 4120575

minus2

119870

2

1(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

) 120578 (120588

2

)]

(59)

where 1198624= 119862

4(119876 120582 119871) ge 1 Note that in the second last

inequality we have used the fact that

∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119883

2

ℎ

1003816

1003816

1003816

1003816

1003816

119889120585 le sup119861120588(1205850)

1003816

1003816

1003816

1003816

1003816

119883

2

ℎ

1003816

1003816

1003816

1003816

1003816

le 119862

0120588

minus2

∮

119861120588(1205850)

|119883ℎ|

2

119889120585 le 119862

0120588

minus2

(60)

In consideration of the fact that 119903 = 2119876(119876 minus 2) gt 2 119876 ge 4

and the assumptions 120579 isin (0 14] and Φ le 1 it follows that

[∮

1198612120579120588(1205850)

(1 + 119906

119903

+ |119883119906|

2

) 119889120585]

2(1minus1119903)

le 119862[∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

119889120585]

2(1minus1119903)

+ 119862(∮

1198612120579120588(1205850)

|119883119906|

2

119889120585)

119903minus1

+ (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

4(1minus1119903)

)

10 Abstract and Applied Analysis

le 119862 [(2120579)

minus2119876(1minus1119903)

Φ(120585

0 120588 119901

0)

2(1minus1119903)

+(2120579)

minus119876(119903minus1)

Φ(120585

0 120588 119901

0)

119903minus1

]

+ (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

2(1minus1119903)

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

4(1minus1119903)

)

le 119862(2120579)

minus119876(119903minus1)

Φ(120585

0 120588 119901

0)

2(1minus1119903)

+ (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

2(1minus1119903)

)

2

(61)

Let 119875 = 1199010+120590(119883ℎ)

12058502120579120588with 119901

0= (119883119906)

12058502120579120588 Combining

these estimates (57)ndash(61) and considering the small condition(51) (it implies 120588 le 120588

1(|119906

12058502120579120588| |119875|) see (64) and (65)) we

deduce that

Φ(120585

0 120579120588) le

1003816

1003816

1003816

1003816

1003816

119861

120579120588(120585

0)

1003816

1003816

1003816

1003816

1003816

minus1

H119899int

1198612120579120588(1205850)

|119883119906 minus 119875|

2

119889120585

le 119862

119888

2

119876

(120579120588)

2∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus 119906

12058502120579120588

minus (119901

0+ 120590(119883ℎ)

12058502120579120588)

times (120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

2

119889120585

+ 2

119876

119862

119888119870

1(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119901

0+ 120590(119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

)

+ 119862

119888

(2120579120588)

2119876(1minus1119903)

(120579120588)

119876

times[∮

1198612120579120588(1205850)

(1 + 119906

119903

+ |119883119906|

2

) 119889120585]

2(1minus1119903)

le 2

119876

119862

4119862

119888(120579

minus119876minus2

120576 + 120579

2

)

times [Φ (120585

0 120588) + 4120575

minus2

119870

2

1

times (

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

) ]

+ 2

119876

119862

119888119870

1(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

+120590(119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

)

+ [2

119876

119862

119888(2120579)

2minus119876(119903minus1)

Φ(120585

0 120588)

2(1minus1119903)

+(1 +

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

2(1minus1119903)

)

2

] 120588

2

(62)

Wenow specify 120576 = 120579119876+4 120579 isin (0 14] such that 2119876+11198624119862

119888120579

2

le

120579

2120591 Note that the small condition (50) implies 12059021198625le 1 with

119862

5= max119862

0 119862

1198882

119876

(2120579)

minus(1198762+4)(119876minus2)

and it yields

2

119876

119862

119888(2120579)

2minus119876(119903minus1)

Φ(120585

0 120588)

2(1minus1119903)

le 1

(63)1003816

1003816

1003816

1003816

1003816

120590(119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

le 120590 sup1198612120579120588(1205850)

|119883ℎ|

le 120590radic119862

0(∮

119861120588(1205850)

|119883ℎ|

2

119889120585) le 120590radic119862

0le 1

(64)

where we have used the a priori estimate (28) for the A-harmonic function ℎ Furthermore using (19) and recallingthe definition of 120590 and 119862

1 we have

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

le

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588minus 119906

1205850120588

1003816

1003816

1003816

1003816

1003816

le

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+ (2120579)

minus1198762

times (∮

119861120588(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus (119883119906)

1205850120588(120585

1

minus 120585

1

0) minus 119906

1205850120588

1003816

1003816

1003816

1003816

1003816

2

119889120585)

12

le

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+ (2120579)

minus1198762

120588radic119862

119901Φ

12

(120585

0 120588)

le

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+

120590

radic

119862

119901

119862

1(2120579)

1198762

le

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+ 120590radic119862

5le

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+ 1

(65)

Combining these estimates with (62) we have

Φ(120585

0 120579120588) le 120579

2120591

Φ(120585

0 120588)

+ [4120575

minus2

119870

2

1(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

)

+ 2

119876

119862

119888119870

1(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

+120590(119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

)] 120578 (120588

2

)

+ [1 + (1 +

1003816

1003816

1003816

1003816

1003816

(119883119906)

1205850 2120579120588

1003816

1003816

1003816

1003816

1003816

2(1minus1119903)

)

2

] 120578 (120588

2

)

le 120579

2120591

Φ(120585

0 120588) + 119870 (

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

)

+ (2 +

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

)

2(119903minus1)

120578 (120588

2

)

le 120579

2120591

Φ(120585

0 120588) + 119870

lowast

(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

)

(66)

Then the proof of Lemma 5 is complete

Abstract and Applied Analysis 11

For 119879 gt 0 we find Φ0(119879) gt 0 (depending on 119876119873 120582 119871

120591 and 120596) such that

120596

2

(2119879 2Φ

0(119879)) + 2Φ

0(119879) le

1

2

120575

2

119862

1Φ

0(119879) le 120579

119876

(1 minus 120579

120591

)

2

119879

2

(67)

With Φ0(119879) from (67) we choose 120588

0(119879) isin (0 1] (depending

on 119876119873 120582 119871 120591 120596 120578 and 120581) such that

120588

0(119879) le 120588

1(1 + 2119879 1 + 2119879)

119862

3119870

2

1(2119879 2119879) 120578 (120588

0(119879)

2

) le 120575

2

119870

0(119879) 120578 (120588

0(119879)

2

) le (120579

2120574

minus 120579

2120591

)Φ

0(119879)

2 (1 + 119862

119901)119870

0(119879)119867 (120588

0(119879)

2

) le 120579

119876

(1 minus 120579

120574

)

2

(120579

2120574

minus 120579

2120591

) 119879

2

(68)

where1198700(119879) = 119870

lowast

(2119879 2119879)By the proof method of of Lemma 51 in [12] and

conditions (67)-(68) Lemma 6 can be proved As is wellknown it is sufficient to complete the proof ofTheorem 1 oncewe obtain Lemma 6

Lemma 6 Assume that for some 1198790gt 0 and 119861

120588(120585

0) subsub Ω one

has

(1) |1199061205850120588| + |(119883119906)

1205850120588| le 119879

0

(2) 120588 le 1205880(119879

0)

(3) Φ(1205850 120588) le Φ

0(119879

0)

Then the small conditions (49)ndash(51) are satisfied on the balls119861

120579119895120588(120585

0) for 119895 isin 119873 cup 0 Moreover the limit Λ

1205850=

lim119895rarrinfin

(119883119906)

1205850120579119895120588exists and the inequality

∮

119861120588(1205850)

1003816

1003816

1003816

1003816

1003816

119883119906 minus Λ

1205850

1003816

1003816

1003816

1003816

1003816

2

119889120585 le 119862

6((

119903

120588

)

2120591

Φ(120585

0 120588) + 119867 (119903

2

))

(69)

is valid for 0 lt 119903 le 120588 with a constant 1198626= 119862

6(119876 119873 120582 119871

120591 119886119899119889 119879

0)

Proof The proof is very similar to the proof of Lemma 51 in[12] We omit it here

Acknowledgments

The project was supported by the National Natural ScienceFoundation of China (no 11201081 and no 11126294) andby the Science and Technology Planning Project of JiangxiProvince China no GJJ13657

References

[1] J Wang and P Niu ldquoOptimal partial regularity for weaksolutions of nonlinear sub-elliptic systems in Carnot groupsrdquoNonlinear Analysis Theory Methods amp Applications vol 72 no11 pp 4162ndash4187 2010

[2] J Wang and D Liao ldquoOptimal partial regularity for sub-elliptic systems with sub-quadratic growth in Carnot groupsrdquoNonlinear Analysis Theory Methods amp Applications vol 75 no4 pp 2499ndash2519 2012

[3] E de Giorgi ldquoUn esempio di estremali discontinue per unproblema variazionale di tipo ellitticordquo Bollettino della UnioneMatematica Italiana vol 4 pp 135ndash137 1968

[4] M Giaquinta Multiple Integrals in the Calculus of Variationsand Nonlinear Elliptic Systems Princeton University PressPrinceton NJ USA 1983

[5] Y Chen and L Wu Second Order Elliptic Equations and EllipticSystems Science Press Beijing China 2003

[6] F Duzaar and K Steffen ldquoOptimal interior and boundary reg-ularity for almost minimizers to elliptic variational integralsrdquoJournal fur die Reine und Angewandte Mathematik vol 546 pp73ndash138 2002

[7] F Duzaar and J F Grotowski ldquoOptimal interior partial regu-larity for nonlinear elliptic systems the method of A-harmonicapproximationrdquo Manuscripta Mathematica vol 103 no 3 pp267ndash298 2000

[8] F Duzaar J F Grotowski and M Kronz ldquoRegularity ofalmost minimizers of quasi-convex variational integrals withsubquadratic growthrdquo Annali di Matematica Pura ed ApplicataIV vol 184 no 4 pp 421ndash448 2005

[9] F Duzaar and G Mingione ldquoThe p-harmonic approximationand the regularity of p-harmonic mapsrdquo Calculus of Variationsand Partial Differential Equations vol 20 no 3 pp 235ndash2562004

[10] F Duzaar and G Mingione ldquoRegularity for degenerate ellipticproblems via p-harmonic approximationrdquo Annales de lrsquoInstitutHenri Poincare Analyse Non Lineaire vol 21 no 5 pp 735ndash7662004

[11] S Chen andZ Tan ldquoThemethod ofA-harmonic approximationand optimal interior partial regularity for nonlinear ellipticsystems under the controllable growth conditionrdquo Journal ofMathematical Analysis and Applications vol 335 no 1 pp 20ndash42 2007

[12] F Duzaar and A Gastel ldquoNonlinear elliptic systems with Dinicontinuous coefficientsrdquo Archiv der Mathematik vol 78 no 1pp 58ndash73 2002

[13] F Duzaar A Gastel and G Mingione ldquoElliptic systemssingular sets and Dini continuityrdquo Communications in PartialDifferential Equations vol 29 no 7-8 pp 1215ndash1240 2004

[14] Y Qiu and Z Tan ldquoOptimal interior partial regularity fornonlinear elliptic systems with Dini continuous coefficientsrdquoActa Mathematica Scientia B vol 30 no 5 pp 1541ndash1554 2010

[15] Y Qiu ldquoOptimal partial regularity of second order nonlin-ear elliptic systems with Dini continuous coefficients for thesuperquadratic caserdquo Nonlinear Analysis Theory Methods ampApplications vol 75 no 8 pp 3574ndash3590 2012

[16] L Capogna ldquoRegularity of quasi-linear equations in theHeisen-berg grouprdquoCommunications on Pure andAppliedMathematicsvol 50 no 9 pp 867ndash889 1997

[17] L Capogna ldquoRegularity for quasilinear equations and 1-quasiconformal maps in Carnot groupsrdquo MathematischeAnnalen vol 313 no 2 pp 263ndash295 1999

[18] S Marchi ldquo1198621120572 local regularity for the solutions of the p-Laplacian on the Heisenberg group for 2 lt 119901 lt 1 +

radic

5rdquoZeitschrift fur Analysis und ihre Anwendungen vol 20 no 3 pp617ndash636 2001

12 Abstract and Applied Analysis

[19] S Marchi ldquo1198621120572 local regularity for the solutions of the p-Laplacian on the Heisenberg groupThe case 1+1radic5 lt 119901 le 2rdquoCommentationes Mathematicae Universitatis Carolinae vol 44no 1 pp 33ndash56 2003

[20] S Marchi ldquoL119901 regularity of the derivative in the secondcommutator direction for nonlinear elliptic equations on theHeisenberg grouprdquo Accademia Nazionale delle Scienze detta deiXL Rendiconti Serie V Memorie di Matematica e ApplicazioniParte I vol 26 pp 1ndash15 2002

[21] A Domokos ldquoDifferentiability of solutions for the non-degenerate p-Laplacian in the Heisenberg grouprdquo Journal ofDifferential Equations vol 204 no 2 pp 439ndash470 2004

[22] A Domokos On the regularity of p-harmonic functions inthe Heisenberg group [PhD thesis] University of PittsburghPittsburgh Pa USA 2004

[23] J J Manfredi and G Mingione ldquoRegularity results for quasilin-ear elliptic equations in the Heisenberg grouprdquo MathematischeAnnalen vol 339 no 3 pp 485ndash544 2007

[24] G Mingione A Zatorska-Goldstein and X Zhong ldquoGradientregularity for elliptic equations in the Heisenberg grouprdquoAdvances in Mathematics vol 222 no 1 pp 62ndash129 2009

[25] L Capogna and N Garofalo ldquoRegularity of minimizers of thecalculus of variations inCarnot groups via hypoellipticity of sys-tems ofHormander typerdquo Journal of the EuropeanMathematicalSociety vol 5 no 1 pp 1ndash40 2003

[26] E Shores ldquoHypoellipticity forlinear degenerate elliptic systemsin Carnot groups and applicationsrdquo httparxivorgabsmath0502569

[27] A Foglein ldquoPartial regularity results for subelliptic systemsin the Heisenberg grouprdquo Calculus of Variations and PartialDifferential Equations vol 32 no 1 pp 25ndash51 2008

[28] G Lu ldquoThe sharp Poincare inequality for free vector fields anendpoint resultrdquo Revista Matematica Iberoamericana vol 10no 2 pp 453ndash466 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 5: Research Article Partial Regularity for Nonlinear Subelliptic Systems …downloads.hindawi.com/journals/aaa/2013/950134.pdf · 2019-07-31 · Partial Regularity for Nonlinear Subelliptic

Abstract and Applied Analysis 5

Adding this to the equations

minus int

119861119877(1205850)

119860

120572

119894(120585 119906 119901

0) 120601

2

(119883119906 minus 119901

0) 119889120585

= 2int

119861119877(1205850)

120601119883120601119860

120572

119894(120585 119906 119901

0) V119889120585

minus int

119861119877(1205850)

119860

120572

119894(120585 119906 119901

0)119883120593

120572

119889120585

0 = int

119861119877(1205850)

119860

120572

119894(120585

0 119906

0 119901

0)119883120593

120572

(32)

It follows that by using the hypotheses (H1) (H3) (ie (23)(21) resp) and (H4)

int

119861119877(1205850)

[119860

120572

119894(120585 119906 119883119906)

minus119860

120572

119894(120585 119906 119901

0)] 120601

2

(119883119906 minus 119901

0) 119889120585

= 2int

119861119877(1205850)

[119860

120572

119894(120585 119906 119901

0)

minus119860

120572

119894(120585 119906 119883119906)] 120601V119883120601119889120585

+ int

119861119877(1205850)

[119860

120572

119894(120585 119906

0+ (120585

1

minus 120585

1

0) 119901

0 119901

0)

minus119860

120572

119894(120585 119906 119901

0) ]119883120593

120572

119889120585

+ int

119861119877(1205850)

[119860

120572

119894(120585

0 119906

0 119901

0)

minus119860

120572

119894(120585 119906

0+ (120585

1

minus 120585

1

0) 119901

0 119901

0)]119883120593

120572

119889120585

+ int

119861119877(1205850)

119861

120572

(120585 119906 119883119906) 120593

120572

119889120585

le 119868 + 119868119868 + 119868119868119868 + 119868119881 + 119881

(33)

where

119868 = 2119871int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

120601

1003816

1003816

1003816

1003816

|V| 1003816100381610038161003816

119883120601

1003816

1003816

1003816

1003816

119889120585

119868119868 = (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

) 120581 (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+ 119877

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

times int

119861119877(1205850)

radic120578 (|V|2) 100381610038161003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

120601

2

119889120585

119868119868119868 = 2 (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

) 120581 (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+ 119877

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

times int

119861119877(1205850)

radic120578 (|V|2) |V| 100381610038161003816

1003816

119883120601

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

120601

1003816

1003816

1003816

1003816

119889120585

119868119881 = (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

) 120581 (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+ 119877

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

times int

119861119877(1205850)

radic120578 (119877

2

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

2

)) [

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

120601

2

+2

1003816

1003816

1003816

1003816

120601

1003816

1003816

1003816

1003816

|V| 1003816100381610038161003816

119883120601

1003816

1003816

1003816

1003816

] 119889120585

119881 = 119862int

119861119877(1205850)

(1 + |119906|

119903minus1

+ |119883119906|

2(1minus1119903)

) 120593119889120585

(34)

Applying (H2) the left hand side of (33) can be estimated as

120582int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

120601

2

119889120585

le int

119861119877(1205850)

[119860

120572

119894(120585 119906 119883119906) minus 119860

120572

119894(120585 119906 119901

0)] 120601

2

(119883119906 minus 119901

0) 119889120585

(35)

For 120576 gt 0 to be fixed later we have using Youngrsquos inequality

119868 le 120576int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

21003816

1003816

1003816

1003816

120601

1003816

1003816

1003816

1003816

2

119889120585 +

119862119871

2

(119877 minus 120588)

2int

119861119877(1205850)

|V|2119889120585

(36)

Using Jensenrsquos inequality (20) and the fact that 120578(119905119904) le 119905120578(119904)for 119905 ge 1 we arrive at

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

120581

2

(sdot) int

119861119877(1205850)

120578 (|V|2) 119889120585

le 120596

119899119877

119876minus2

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

120581

2

(sdot) 119877

2

120578 (∮

119861119877(1205850)

|V|2119889120585)

le 120596

119899119877

119876minus2

[∮

119861119877(1205850)

|V|2119889120585

+ (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

120581

2

(sdot) 119877

2

120578

times ((1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

120581

2

(sdot) 119877

2

) ]

le 119877

minus2

int

119861119877(1205850)

|V|2119889120585

+ 120596

119899119877

119876

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

4

120581

4

(sdot) 120578 (119877

2

)

(37)

where 120581(sdot) is an abbreviation of the function 120581(|1199060| + |119901

0|)

Also note that the application of (20) in the second lastinequality is possible by our choice 119877 le 120588

1(|119906

0| + |119901

0|)

6 Abstract and Applied Analysis

Using Youngrsquos inequality and (37) in 119868119868 we obtain

119868119868 le 120576int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

21003816

1003816

1003816

1003816

120601

1003816

1003816

1003816

1003816

2

119889120585

+ 120576

minus1

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

120581

2

(sdot) int

119861119877(1205850)

120578 (|V|2) 119889120585

le 120576int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

21003816

1003816

1003816

1003816

120601

1003816

1003816

1003816

1003816

2

119889120585

+

1

120576(119877 minus 120588)

2int

119861119877(1205850)

|V|2119889120585

+ 120576

minus1

120596

119899119877

119876

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

4

120581

4

(sdot) 120578 (119877

2

)

(38)

And similarly we see

119868119868119868 le

4119862

(119877 minus 120588)

2int

119861119877(1205850)

|V|2119889120585

+ (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

120581

2

(sdot) int

119861119877(1205850)

120578 (|V|2) 119889120585

le

119862

(119877 minus 120588)

2int

119861119877(1205850)

|V|2119889120585

+ 120596

119899119877

119876

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

4

120581

4

(sdot) 120578 (119877

2

)

119868119881 le 120576int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

21003816

1003816

1003816

1003816

120601

1003816

1003816

1003816

1003816

2

119889120585

+

4119862120576

(119877 minus 120588)

2int

119861119877(1205850)

|V|2119889120585

+ 120576

minus1

120596

119899119877

119876

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

120581

2

(sdot) 120578

times (∮

119861119877(1205850)

119877

2

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

2

) 119889120585)

le 120576int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

21003816

1003816

1003816

1003816

120601

1003816

1003816

1003816

1003816

2

119889120585

+

119862120576

(119877 minus 120588)

2int

119861119877(1205850)

|V|2119889120585

+ 120576

minus1

120596

119899119877

119876

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

4

120581

4

(sdot) 120578 (119877

2

)

(39)

Here we have used 120581 ge 1 in the last inequality

By Holderrsquos inequality (19) and Youngrsquos inequality onegets

119881 le 119862(int

119861119877(1205850)

1003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119903

119889120585)

1119903

times (int

119861119877(1205850)

(1 + |119906|

119903

+ |119883119906|

2

) 119889120585)

(119903minus1)119903

le 119862(int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

2

119889120585)

12

times (int

119861119877(1205850)

(1 + |119906|

119903

+ |119883119906|

2

) 119889120585)

(119903minus1)119903

le 120576int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

2

119889120585 + 119862 (120576)

times (int

119861119877(1205850)

(1 + |119906|

119903

+ |119883119906|

2

) 119889120585)

2(119903minus1)119903

le 120576int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

21003816

1003816

1003816

1003816

120601

1003816

1003816

1003816

1003816

2

119889120585

+

119862120576

(119877 minus 120588)

2int

119861119877(1205850)

|V|2119889120585

+ 119862 (120576) (int

119861119877(1205850)

(1 + |119906|

119903

+ |119883119906|

2

) 119889120585)

2(119903minus1)119903

(40)

where we have used the fact that |119883120593| le 120601|119883119906 minus 1199010| + 119862(119877 minus

120588)|V|Applying these estimates to (37) we obtain

(120582 minus 4120576) int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

120601

2

119889120585

le

119862 (119871 120576)

(119877 minus 120588)

2int

119861119877(1205850)

|V|2119889120585

+ (120576

minus1

+ 2)120596

119899119877

119876

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

4

120581

4

(sdot) 120578 (119877

2

)

+ 119862 (120576) (int

119861119877(1205850)

(1 + |119906|

119903

+ |119883119906|

2

) 119889120585)

2(119903minus1)119903

(41)

Choosing 120576 = 1205828 we obtain the desired inequality (29)

4 Proof of the Main Theorem

In this section we will complete the proof of the partialregularity results via the following lemmas In the sequel wealways suppose that 119906 isin 11986711988212(ΩR119873) is a weak solution to(1) with the assumptions of (H1)ndash(H4) and (1205831)ndash(1205833)

Abstract and Applied Analysis 7

Lemma 4 Let 119861120588(120585

0) subsub Ω with 120588 le 120588

1(|119906

0| |119901

0|) and 120593 isin

119862

infin

0(119861

120588(120585

0)R119873) satisfying |120593| le 120588

2 and sup119861120588(1205850)

|119883120593| le 1Then there exists a constant 119862

1ge 1 such that

∮

119861120588(1205850)

119860

120572

119894119901119895

120573

(120585

0 119906

0 119901

0) (119883119906 minus 119901

0)119883120593

120572

119889120585

le 119862

1[Φ (120585

0 120588 119901

0)

+ 120596 (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

Φ (120585

0 120588 119901

0))Φ

12

(120585

0 120588 119901

0)

+ 119870

1(

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)radic120578 (120588

2

)] sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

(42)

Proof Using the fact that int119861120588(1205850)

119860

120572

119894(120585

0 119906

0 119901

0)119883120593

120572

119889120585 = 0 andthe weak form (8) we deduce

∮

119861120588(1205850)

[int

1

0

119860

120572

119894119901119895

120573

(120585

0 119906

0 120579119883119906 + (1 minus 120579) 119901

0)

times (119883119906 minus 119901

0) 119889120579]119883120593

120572

119889120585

= ∮

119861120588(1205850)

[119860

120572

119894(120585

0 119906

0 119883119906)

minus119860

120572

119894(120585

0 119906

0 119901

0)]119883120593

120572

119889120585

= ∮

119861120588(1205850)

[119860

120572

119894(120585

0 119906

0 119883119906)

minus119860

120572

119894(120585 119906 119883119906)]119883120593

120572

119889120585

+ ∮

119861120588(1205850)

119861

120572

(120585 119906 119883119906) 120593

120572

119889120585

(43)

It yields

∮

119861120588(1205850)

119860

120572

119894119901119895

120573

(120585

0 119906

0 119901

0) (119883119906 minus 119901

0)119883120593

120572

119889120585

= ∮

119861120588(1205850)

[int

1

0

(119860

120572

119894119901119895

120573

(120585

0 119906

0 119901

0)

minus119860

120572

119894119901119895

120573

(120585

0 119906

0 120579119883119906 + (1 minus 120579) 119901

0))

times (119883119906 minus 119901

0) 119889120579] 119889120585 sup

119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

+ ∮

119861120588(1205850)

[119860

120572

119894(120585

0 119906

0 119883119906)

minus119860

120572

119894(120585 119906

0+ 119901

0(120585 minus 120585

0) 119883119906)] sup

119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

+ ∮

119861120588(1205850)

[119860

120572

119894(120585 119906

0+ 119901

0(120585 minus 120585

0) 119883119906)

minus119860

120572

119894(120585 119906 119883119906)] sup

119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

+ ∮

119861120588(1205850)

119861

120572

(120585 119906 119883119906) 120593

120572

119889120585

= 119868

1015840

+ 119868119868

1015840

+ 119868119868119868

1015840

+ 119868119881

1015840

(44)

Using (22) Holderrsquos inequality the fact that 119905 rarr 120596

2

(119904 119905) isconcave and Jensenrsquos inequality we have

119868

1015840

le sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

∮

119861120588(1205850)

120596 (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

119889120585

le sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

[∮

119861120588(1205850)

120596

2

(

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

) 119889120585]

12

times[∮

119861120588(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

119889120585]

12

le 120596(

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

Φ (120585

0 120588 119901

0))Φ

12

(120585

0 120588 119901

0) sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

(45)

Similarly using (21) and the fact that 120578(119905119904) le 119905120578(119904) for 119905 ge 1we obtain

119868119868

1015840

le sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

120581 (sdot)radic120578 (120588

2

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

2

))

times ∮

119861120588(1205850)

(1 + |119883119906|) 119889120585

le sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

120581 (sdot)radic120578 (120588

2

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

)

times ∮

119861120588(1205850)

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

) 119889120585

le sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

[(∮

119861120588(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

119889120585)

+ 120581

2

(sdot) (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

120578 (120588

2

)

+ 120581 (sdot) (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

3

radic120578 (120588

2

) ]

le [Φ (120585

0 120588 119901

0)+ 2120581

2

(sdot) (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

3

radic120578 (120588

2

)]

times sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

(46)

8 Abstract and Applied Analysis

where we have used the fact that 120578(1205882) le

radic

120578(120588

2

) whichfollows from the nondecreasing property of the function 120578(119905)(1205784) and our assumption 120588 le 120588

1le 1

In the same way it follows that by using (21) (37) and(19)

119868119868119868

1015840

le sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

120581 (sdot)∮

119861120588(1205850)

radic120578 (|V|2) (1 + |119883119906|) 119889120585

le sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

[∮

119861120588(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

119889120585

+ 120581

2

(sdot)∮

119861120588(1205850)

120578 (|V|2) 119889120585

+ 120581 (sdot) (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)∮

119861120588(1205850)

radic120578 (|V|2) 119889120585]

le sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

[Φ (120585

0 120588 119901

0)

+ 2120588

minus2

∮

119861120588(1205850)

|V|2119889120585 + 1205814 (sdot) 120578 (1205882)

+ 120581

2

(sdot) (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

radic120578 (120588

2

) ]

le [(1 + 2119862

119901)Φ (120585

0 120588 119901

0)

+ 2120581

4

(sdot) (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

radic120578 (120588

2

)]

(47)

Using Holderrsquos inequality (19) and Youngrsquos inequality wehave

119868119881

1015840

le 119862∮

119861120588(1205850)

(1 + |119906|

119903minus1

+ |119883119906|

2(1minus1119903)

)

1003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119889120585

le 119862∮

119861120588(1205850)

|119883119906|

2(1minus1119903) 10038161003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119889120585

+ 119862∮

119861120588(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus 119906

0minus 119901

0(120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

119903minus11003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119889120585

+ 119862120588

2

[1 + (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

119903minus1

]

le 119862(∮

119861120588(1205850)

|119883119906|

2

119889120585)

(1minus1119903)

(∮

119861120588(1205850)

1003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119903

119889120585)

1119903

+ 119862(∮

119861120588(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus 119906

0minus 119901

0(120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

119903

119889120585)

(1minus1119903)

times (∮

119861120588(1205850)

1003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119903

119889120585)

1119903

+ 119862120588

2

[1 + (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

119903minus1

]

le 119862(∮

119861120588(1205850)

|119883119906|

2

119889120585)

(1minus1119903)

(∮

119861120588(1205850)

1003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119903

119889120585)

1119903

+ (∮

119861120588(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

119889120585)

(1199032)(1minus1119903)

times (∮

119861120588(1205850)

1003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119903

119889120585)

1119903

+ 119862120588

2

[1 + (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

119903minus1

]

le 119862119906

11986711988212(∮

119861120588(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

119889120585)

(1minus1119903)

times (∮

119861120588(1205850)

1003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119903

119889120585)

1119903

+ 119862120588 [1 + (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

119903minus1

]

le 119862119906

11986711988212(119861120588(1205850))

(∮

119861120588(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

119889120585)

+ 119862120588

2119903

+ 119862120588

2

[1 + (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

119903minus1

]

le 119862

2Φ(120585

0 120588 119901

0) + 119862120588

2

(1 +

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

119903minus1

le 119862

2Φ(120585

0 120588 119901

0) + 119862120581 (sdot) (1 +

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

radic120578 (120588

2

)

(48)

where we have used the assumption (1205784) and the fact that 119903 =2119876(119876 minus 2) = (2119899 + 4)2119899 le 3 and 119862

2= 119862119906

11986711988212(119861120588(1205850))

ge

1 Combining these estimates we obtain the conclusion with119862

1= (1 + 119862

2+ 2119862

119901) ge 1

Lemma 5 Assume that the conditions of Lemma 2 and thefollowing smallness conditions hold

120596 (

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

1003816

(119883119906)

1205850120588

1003816

1003816

1003816

1003816

1003816

Φ (120585

0 120588 (119883119906)

1205850120588))

+ Φ

12

(120585

0 120588 (119883119906)

1205850120588) le

120575

2

(49)

119862

3119870

2

1(

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

1205850120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

) le 120575

2 (50)

with 1198623= 8119862

2

1119862

5 together with

120588 le 120588

1(1 +

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

1 +

1003816

1003816

1003816

1003816

1003816

(119883119906)

1205850120588

1003816

1003816

1003816

1003816

1003816

) (51)

Then the following growth condition holds for 120591 isin [120574 1)

Φ (120585

0 120579120588) le 120579

2120591

Φ(120585

0 120588)

+ 119870

lowast

(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

)

(52)

where one abbreviates Φ(1205850 119903) = Φ(120585

0 119903 (119883119906)

1205850119903) and

119870

lowast

(119904 119905) = 119870(119904 119905) + (2 + 119904 + 119905)

2(119903minus1) with 119870(119904 119905) = (4120575

minus2

+

2

119876

119862

119888)119870

2

1(1 + 119904 1 + 119905)

Proof We define 119908 = [119906 minus 1199061205850120588

minus 119901

0(120585

1

minus 120585

1

0)]120590

minus1 where

120590 = 119862

1radic

Φ(120585

0 120588 119901

0) + 4120575

minus2119870

2

1(

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

) 120578 (120588

2)

(53)

Abstract and Applied Analysis 9

Then we have 119883119908 = 120590

minus1

(119883119906 minus 119901

0) Now we consider

119861

120588(120585

0) subsub Ω such that 120588 le 120588

1(|119906

0| |119901

0|) Applying Lemma 4

on 119861120588(120585

0) to 119906 we have for any 120593 isin 119862infin

0(119861

120588(120585

0)R119873)

∮

119861120588(1205850)

|119883119908|

2

119889120585 = 120590

minus2

Φ(120585

0 120588 119901

0) le

1

119862

2

1

le 1 (54)

∮

119861120588(1205850)

119860

120572

119894119901119895

120573

(120585

0 119906

0 119901

0)119883119908119883120593119889120585

le [Φ

12

(120585

0 120588 119901

0)

+120596 (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

Φ (120585

0 120588 119901

0)) +

120575

2

] sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

(55)

In consideration of the small condition (49) we see that(54) and (55) imply conditions (26) in Lemma 2 Also notethat (H1) and (H3) imply condition (25) So there exists an119860

120572

119894119901119895

120573

(120585

0 119906

0 119901

0)-harmonic function ℎ isin 11986711988212(119861

120588(120585

0)R119873)

such that

∮

119861120588(1205850)

|119883ℎ|

2

119889120585 le 1 120588

minus2

∮

119861120588(1205850)

|119908 minus ℎ|

2

119889120585 le 120576 (56)

Taking 1199060= 119906

12058502120579120588 120579 isin (0 14] and replacing 119901

0by 1199010+

120590(119883ℎ)

1205850 2120579120588 we use Lemma 3 to obtain

int

119861120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119883119906 minus 119901

0minus 120590(119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

2

119889120585

le 119862

119888[

1

(120579120588)

2int

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus 119906

12058502120579120588

minus (119901

0+ 120590(119883ℎ)

12058502120579120588)

times (120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

2

119889120585 + 119865]

(57)

where

119865 = 120596

119899(2120579120588)

119876

119870

1(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119901

0+ 120590(119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 ((2120579120588)

2

)

+ [int

1198612120579120588(1205850)

(1 + 119906

119903

+ |119883119906|

2

) 119889120585]

2(1minus1119903)

(58)

Using the fact that 119906 minus (1199010+ 120590(119883ℎ)

12058502120579120588)(120585

1

minus 120585

1

0) has mean

value 11990612058502120579120588

on the ball 1198612120579120588(120585

0) the definition of119908 and (19)

we have

1

(120579120588)

2∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus 119906

1205850 2120579120588minus (119901

0+ 120590(119883ℎ)

12058502120579120588) (120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

2

119889120585

le

4120590

2

(2120579120588)

2∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119908 minus ℎ

1205850 2120579120588

minus(119883ℎ)

12058502120579120588(120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

2

119889120585

le

4120590

2

(2120579120588)

2[∮

1198612120579120588(1205850)

|119908 minus ℎ|

2

119889120585

+ ∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

ℎ minus ℎ

1205850 2120579120588

minus(119883ℎ)

12058502120579120588(120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

2

119889120585]

le 4120590

2

[(2120579)

minus119876minus2

120576 + 119862

119901∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119883ℎ minus (119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

2

119889120585]

le 4120590

2

[(2120579)

minus119876minus2

120576 + 119862

2

119901(2120579120588)

2

∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119883

2

ℎ

1003816

1003816

1003816

1003816

1003816

2

119889120585]

le 4120590

2

[(2120579)

minus119876minus2

120576 + 119862

2

119901(2120579)

2

119862

0]

le 119862

4(120579

minus119876minus2

120576 + 120579

2

) [Φ (120585

0 120588 119901

0)

+ 4120575

minus2

119870

2

1(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

) 120578 (120588

2

)]

(59)

where 1198624= 119862

4(119876 120582 119871) ge 1 Note that in the second last

inequality we have used the fact that

∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119883

2

ℎ

1003816

1003816

1003816

1003816

1003816

119889120585 le sup119861120588(1205850)

1003816

1003816

1003816

1003816

1003816

119883

2

ℎ

1003816

1003816

1003816

1003816

1003816

le 119862

0120588

minus2

∮

119861120588(1205850)

|119883ℎ|

2

119889120585 le 119862

0120588

minus2

(60)

In consideration of the fact that 119903 = 2119876(119876 minus 2) gt 2 119876 ge 4

and the assumptions 120579 isin (0 14] and Φ le 1 it follows that

[∮

1198612120579120588(1205850)

(1 + 119906

119903

+ |119883119906|

2

) 119889120585]

2(1minus1119903)

le 119862[∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

119889120585]

2(1minus1119903)

+ 119862(∮

1198612120579120588(1205850)

|119883119906|

2

119889120585)

119903minus1

+ (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

4(1minus1119903)

)

10 Abstract and Applied Analysis

le 119862 [(2120579)

minus2119876(1minus1119903)

Φ(120585

0 120588 119901

0)

2(1minus1119903)

+(2120579)

minus119876(119903minus1)

Φ(120585

0 120588 119901

0)

119903minus1

]

+ (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

2(1minus1119903)

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

4(1minus1119903)

)

le 119862(2120579)

minus119876(119903minus1)

Φ(120585

0 120588 119901

0)

2(1minus1119903)

+ (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

2(1minus1119903)

)

2

(61)

Let 119875 = 1199010+120590(119883ℎ)

12058502120579120588with 119901

0= (119883119906)

12058502120579120588 Combining

these estimates (57)ndash(61) and considering the small condition(51) (it implies 120588 le 120588

1(|119906

12058502120579120588| |119875|) see (64) and (65)) we

deduce that

Φ(120585

0 120579120588) le

1003816

1003816

1003816

1003816

1003816

119861

120579120588(120585

0)

1003816

1003816

1003816

1003816

1003816

minus1

H119899int

1198612120579120588(1205850)

|119883119906 minus 119875|

2

119889120585

le 119862

119888

2

119876

(120579120588)

2∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus 119906

12058502120579120588

minus (119901

0+ 120590(119883ℎ)

12058502120579120588)

times (120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

2

119889120585

+ 2

119876

119862

119888119870

1(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119901

0+ 120590(119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

)

+ 119862

119888

(2120579120588)

2119876(1minus1119903)

(120579120588)

119876

times[∮

1198612120579120588(1205850)

(1 + 119906

119903

+ |119883119906|

2

) 119889120585]

2(1minus1119903)

le 2

119876

119862

4119862

119888(120579

minus119876minus2

120576 + 120579

2

)

times [Φ (120585

0 120588) + 4120575

minus2

119870

2

1

times (

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

) ]

+ 2

119876

119862

119888119870

1(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

+120590(119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

)

+ [2

119876

119862

119888(2120579)

2minus119876(119903minus1)

Φ(120585

0 120588)

2(1minus1119903)

+(1 +

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

2(1minus1119903)

)

2

] 120588

2

(62)

Wenow specify 120576 = 120579119876+4 120579 isin (0 14] such that 2119876+11198624119862

119888120579

2

le

120579

2120591 Note that the small condition (50) implies 12059021198625le 1 with

119862

5= max119862

0 119862

1198882

119876

(2120579)

minus(1198762+4)(119876minus2)

and it yields

2

119876

119862

119888(2120579)

2minus119876(119903minus1)

Φ(120585

0 120588)

2(1minus1119903)

le 1

(63)1003816

1003816

1003816

1003816

1003816

120590(119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

le 120590 sup1198612120579120588(1205850)

|119883ℎ|

le 120590radic119862

0(∮

119861120588(1205850)

|119883ℎ|

2

119889120585) le 120590radic119862

0le 1

(64)

where we have used the a priori estimate (28) for the A-harmonic function ℎ Furthermore using (19) and recallingthe definition of 120590 and 119862

1 we have

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

le

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588minus 119906

1205850120588

1003816

1003816

1003816

1003816

1003816

le

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+ (2120579)

minus1198762

times (∮

119861120588(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus (119883119906)

1205850120588(120585

1

minus 120585

1

0) minus 119906

1205850120588

1003816

1003816

1003816

1003816

1003816

2

119889120585)

12

le

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+ (2120579)

minus1198762

120588radic119862

119901Φ

12

(120585

0 120588)

le

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+

120590

radic

119862

119901

119862

1(2120579)

1198762

le

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+ 120590radic119862

5le

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+ 1

(65)

Combining these estimates with (62) we have

Φ(120585

0 120579120588) le 120579

2120591

Φ(120585

0 120588)

+ [4120575

minus2

119870

2

1(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

)

+ 2

119876

119862

119888119870

1(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

+120590(119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

)] 120578 (120588

2

)

+ [1 + (1 +

1003816

1003816

1003816

1003816

1003816

(119883119906)

1205850 2120579120588

1003816

1003816

1003816

1003816

1003816

2(1minus1119903)

)

2

] 120578 (120588

2

)

le 120579

2120591

Φ(120585

0 120588) + 119870 (

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

)

+ (2 +

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

)

2(119903minus1)

120578 (120588

2

)

le 120579

2120591

Φ(120585

0 120588) + 119870

lowast

(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

)

(66)

Then the proof of Lemma 5 is complete

Abstract and Applied Analysis 11

For 119879 gt 0 we find Φ0(119879) gt 0 (depending on 119876119873 120582 119871

120591 and 120596) such that

120596

2

(2119879 2Φ

0(119879)) + 2Φ

0(119879) le

1

2

120575

2

119862

1Φ

0(119879) le 120579

119876

(1 minus 120579

120591

)

2

119879

2

(67)

With Φ0(119879) from (67) we choose 120588

0(119879) isin (0 1] (depending

on 119876119873 120582 119871 120591 120596 120578 and 120581) such that

120588

0(119879) le 120588

1(1 + 2119879 1 + 2119879)

119862

3119870

2

1(2119879 2119879) 120578 (120588

0(119879)

2

) le 120575

2

119870

0(119879) 120578 (120588

0(119879)

2

) le (120579

2120574

minus 120579

2120591

)Φ

0(119879)

2 (1 + 119862

119901)119870

0(119879)119867 (120588

0(119879)

2

) le 120579

119876

(1 minus 120579

120574

)

2

(120579

2120574

minus 120579

2120591

) 119879

2

(68)

where1198700(119879) = 119870

lowast

(2119879 2119879)By the proof method of of Lemma 51 in [12] and

conditions (67)-(68) Lemma 6 can be proved As is wellknown it is sufficient to complete the proof ofTheorem 1 oncewe obtain Lemma 6

Lemma 6 Assume that for some 1198790gt 0 and 119861

120588(120585

0) subsub Ω one

has

(1) |1199061205850120588| + |(119883119906)

1205850120588| le 119879

0

(2) 120588 le 1205880(119879

0)

(3) Φ(1205850 120588) le Φ

0(119879

0)

Then the small conditions (49)ndash(51) are satisfied on the balls119861

120579119895120588(120585

0) for 119895 isin 119873 cup 0 Moreover the limit Λ

1205850=

lim119895rarrinfin

(119883119906)

1205850120579119895120588exists and the inequality

∮

119861120588(1205850)

1003816

1003816

1003816

1003816

1003816

119883119906 minus Λ

1205850

1003816

1003816

1003816

1003816

1003816

2

119889120585 le 119862

6((

119903

120588

)

2120591

Φ(120585

0 120588) + 119867 (119903

2

))

(69)

is valid for 0 lt 119903 le 120588 with a constant 1198626= 119862

6(119876 119873 120582 119871

120591 119886119899119889 119879

0)

Proof The proof is very similar to the proof of Lemma 51 in[12] We omit it here

Acknowledgments

The project was supported by the National Natural ScienceFoundation of China (no 11201081 and no 11126294) andby the Science and Technology Planning Project of JiangxiProvince China no GJJ13657

References

[1] J Wang and P Niu ldquoOptimal partial regularity for weaksolutions of nonlinear sub-elliptic systems in Carnot groupsrdquoNonlinear Analysis Theory Methods amp Applications vol 72 no11 pp 4162ndash4187 2010

[2] J Wang and D Liao ldquoOptimal partial regularity for sub-elliptic systems with sub-quadratic growth in Carnot groupsrdquoNonlinear Analysis Theory Methods amp Applications vol 75 no4 pp 2499ndash2519 2012

[3] E de Giorgi ldquoUn esempio di estremali discontinue per unproblema variazionale di tipo ellitticordquo Bollettino della UnioneMatematica Italiana vol 4 pp 135ndash137 1968

[4] M Giaquinta Multiple Integrals in the Calculus of Variationsand Nonlinear Elliptic Systems Princeton University PressPrinceton NJ USA 1983

[5] Y Chen and L Wu Second Order Elliptic Equations and EllipticSystems Science Press Beijing China 2003

[6] F Duzaar and K Steffen ldquoOptimal interior and boundary reg-ularity for almost minimizers to elliptic variational integralsrdquoJournal fur die Reine und Angewandte Mathematik vol 546 pp73ndash138 2002

[7] F Duzaar and J F Grotowski ldquoOptimal interior partial regu-larity for nonlinear elliptic systems the method of A-harmonicapproximationrdquo Manuscripta Mathematica vol 103 no 3 pp267ndash298 2000

[8] F Duzaar J F Grotowski and M Kronz ldquoRegularity ofalmost minimizers of quasi-convex variational integrals withsubquadratic growthrdquo Annali di Matematica Pura ed ApplicataIV vol 184 no 4 pp 421ndash448 2005

[9] F Duzaar and G Mingione ldquoThe p-harmonic approximationand the regularity of p-harmonic mapsrdquo Calculus of Variationsand Partial Differential Equations vol 20 no 3 pp 235ndash2562004

[10] F Duzaar and G Mingione ldquoRegularity for degenerate ellipticproblems via p-harmonic approximationrdquo Annales de lrsquoInstitutHenri Poincare Analyse Non Lineaire vol 21 no 5 pp 735ndash7662004

[11] S Chen andZ Tan ldquoThemethod ofA-harmonic approximationand optimal interior partial regularity for nonlinear ellipticsystems under the controllable growth conditionrdquo Journal ofMathematical Analysis and Applications vol 335 no 1 pp 20ndash42 2007

[12] F Duzaar and A Gastel ldquoNonlinear elliptic systems with Dinicontinuous coefficientsrdquo Archiv der Mathematik vol 78 no 1pp 58ndash73 2002

[13] F Duzaar A Gastel and G Mingione ldquoElliptic systemssingular sets and Dini continuityrdquo Communications in PartialDifferential Equations vol 29 no 7-8 pp 1215ndash1240 2004

[14] Y Qiu and Z Tan ldquoOptimal interior partial regularity fornonlinear elliptic systems with Dini continuous coefficientsrdquoActa Mathematica Scientia B vol 30 no 5 pp 1541ndash1554 2010

[15] Y Qiu ldquoOptimal partial regularity of second order nonlin-ear elliptic systems with Dini continuous coefficients for thesuperquadratic caserdquo Nonlinear Analysis Theory Methods ampApplications vol 75 no 8 pp 3574ndash3590 2012

[16] L Capogna ldquoRegularity of quasi-linear equations in theHeisen-berg grouprdquoCommunications on Pure andAppliedMathematicsvol 50 no 9 pp 867ndash889 1997

[17] L Capogna ldquoRegularity for quasilinear equations and 1-quasiconformal maps in Carnot groupsrdquo MathematischeAnnalen vol 313 no 2 pp 263ndash295 1999

[18] S Marchi ldquo1198621120572 local regularity for the solutions of the p-Laplacian on the Heisenberg group for 2 lt 119901 lt 1 +

radic

5rdquoZeitschrift fur Analysis und ihre Anwendungen vol 20 no 3 pp617ndash636 2001

12 Abstract and Applied Analysis

[19] S Marchi ldquo1198621120572 local regularity for the solutions of the p-Laplacian on the Heisenberg groupThe case 1+1radic5 lt 119901 le 2rdquoCommentationes Mathematicae Universitatis Carolinae vol 44no 1 pp 33ndash56 2003

[20] S Marchi ldquoL119901 regularity of the derivative in the secondcommutator direction for nonlinear elliptic equations on theHeisenberg grouprdquo Accademia Nazionale delle Scienze detta deiXL Rendiconti Serie V Memorie di Matematica e ApplicazioniParte I vol 26 pp 1ndash15 2002

[21] A Domokos ldquoDifferentiability of solutions for the non-degenerate p-Laplacian in the Heisenberg grouprdquo Journal ofDifferential Equations vol 204 no 2 pp 439ndash470 2004

[22] A Domokos On the regularity of p-harmonic functions inthe Heisenberg group [PhD thesis] University of PittsburghPittsburgh Pa USA 2004

[23] J J Manfredi and G Mingione ldquoRegularity results for quasilin-ear elliptic equations in the Heisenberg grouprdquo MathematischeAnnalen vol 339 no 3 pp 485ndash544 2007

[24] G Mingione A Zatorska-Goldstein and X Zhong ldquoGradientregularity for elliptic equations in the Heisenberg grouprdquoAdvances in Mathematics vol 222 no 1 pp 62ndash129 2009

[25] L Capogna and N Garofalo ldquoRegularity of minimizers of thecalculus of variations inCarnot groups via hypoellipticity of sys-tems ofHormander typerdquo Journal of the EuropeanMathematicalSociety vol 5 no 1 pp 1ndash40 2003

[26] E Shores ldquoHypoellipticity forlinear degenerate elliptic systemsin Carnot groups and applicationsrdquo httparxivorgabsmath0502569

[27] A Foglein ldquoPartial regularity results for subelliptic systemsin the Heisenberg grouprdquo Calculus of Variations and PartialDifferential Equations vol 32 no 1 pp 25ndash51 2008

[28] G Lu ldquoThe sharp Poincare inequality for free vector fields anendpoint resultrdquo Revista Matematica Iberoamericana vol 10no 2 pp 453ndash466 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 6: Research Article Partial Regularity for Nonlinear Subelliptic Systems …downloads.hindawi.com/journals/aaa/2013/950134.pdf · 2019-07-31 · Partial Regularity for Nonlinear Subelliptic

6 Abstract and Applied Analysis

Using Youngrsquos inequality and (37) in 119868119868 we obtain

119868119868 le 120576int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

21003816

1003816

1003816

1003816

120601

1003816

1003816

1003816

1003816

2

119889120585

+ 120576

minus1

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

120581

2

(sdot) int

119861119877(1205850)

120578 (|V|2) 119889120585

le 120576int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

21003816

1003816

1003816

1003816

120601

1003816

1003816

1003816

1003816

2

119889120585

+

1

120576(119877 minus 120588)

2int

119861119877(1205850)

|V|2119889120585

+ 120576

minus1

120596

119899119877

119876

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

4

120581

4

(sdot) 120578 (119877

2

)

(38)

And similarly we see

119868119868119868 le

4119862

(119877 minus 120588)

2int

119861119877(1205850)

|V|2119889120585

+ (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

120581

2

(sdot) int

119861119877(1205850)

120578 (|V|2) 119889120585

le

119862

(119877 minus 120588)

2int

119861119877(1205850)

|V|2119889120585

+ 120596

119899119877

119876

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

4

120581

4

(sdot) 120578 (119877

2

)

119868119881 le 120576int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

21003816

1003816

1003816

1003816

120601

1003816

1003816

1003816

1003816

2

119889120585

+

4119862120576

(119877 minus 120588)

2int

119861119877(1205850)

|V|2119889120585

+ 120576

minus1

120596

119899119877

119876

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

120581

2

(sdot) 120578

times (∮

119861119877(1205850)

119877

2

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

2

) 119889120585)

le 120576int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

21003816

1003816

1003816

1003816

120601

1003816

1003816

1003816

1003816

2

119889120585

+

119862120576

(119877 minus 120588)

2int

119861119877(1205850)

|V|2119889120585

+ 120576

minus1

120596

119899119877

119876

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

4

120581

4

(sdot) 120578 (119877

2

)

(39)

Here we have used 120581 ge 1 in the last inequality

By Holderrsquos inequality (19) and Youngrsquos inequality onegets

119881 le 119862(int

119861119877(1205850)

1003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119903

119889120585)

1119903

times (int

119861119877(1205850)

(1 + |119906|

119903

+ |119883119906|

2

) 119889120585)

(119903minus1)119903

le 119862(int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

2

119889120585)

12

times (int

119861119877(1205850)

(1 + |119906|

119903

+ |119883119906|

2

) 119889120585)

(119903minus1)119903

le 120576int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

2

119889120585 + 119862 (120576)

times (int

119861119877(1205850)

(1 + |119906|

119903

+ |119883119906|

2

) 119889120585)

2(119903minus1)119903

le 120576int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

21003816

1003816

1003816

1003816

120601

1003816

1003816

1003816

1003816

2

119889120585

+

119862120576

(119877 minus 120588)

2int

119861119877(1205850)

|V|2119889120585

+ 119862 (120576) (int

119861119877(1205850)

(1 + |119906|

119903

+ |119883119906|

2

) 119889120585)

2(119903minus1)119903

(40)

where we have used the fact that |119883120593| le 120601|119883119906 minus 1199010| + 119862(119877 minus

120588)|V|Applying these estimates to (37) we obtain

(120582 minus 4120576) int

119861119877(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

120601

2

119889120585

le

119862 (119871 120576)

(119877 minus 120588)

2int

119861119877(1205850)

|V|2119889120585

+ (120576

minus1

+ 2)120596

119899119877

119876

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

4

120581

4

(sdot) 120578 (119877

2

)

+ 119862 (120576) (int

119861119877(1205850)

(1 + |119906|

119903

+ |119883119906|

2

) 119889120585)

2(119903minus1)119903

(41)

Choosing 120576 = 1205828 we obtain the desired inequality (29)

4 Proof of the Main Theorem

In this section we will complete the proof of the partialregularity results via the following lemmas In the sequel wealways suppose that 119906 isin 11986711988212(ΩR119873) is a weak solution to(1) with the assumptions of (H1)ndash(H4) and (1205831)ndash(1205833)

Abstract and Applied Analysis 7

Lemma 4 Let 119861120588(120585

0) subsub Ω with 120588 le 120588

1(|119906

0| |119901

0|) and 120593 isin

119862

infin

0(119861

120588(120585

0)R119873) satisfying |120593| le 120588

2 and sup119861120588(1205850)

|119883120593| le 1Then there exists a constant 119862

1ge 1 such that

∮

119861120588(1205850)

119860

120572

119894119901119895

120573

(120585

0 119906

0 119901

0) (119883119906 minus 119901

0)119883120593

120572

119889120585

le 119862

1[Φ (120585

0 120588 119901

0)

+ 120596 (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

Φ (120585

0 120588 119901

0))Φ

12

(120585

0 120588 119901

0)

+ 119870

1(

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)radic120578 (120588

2

)] sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

(42)

Proof Using the fact that int119861120588(1205850)

119860

120572

119894(120585

0 119906

0 119901

0)119883120593

120572

119889120585 = 0 andthe weak form (8) we deduce

∮

119861120588(1205850)

[int

1

0

119860

120572

119894119901119895

120573

(120585

0 119906

0 120579119883119906 + (1 minus 120579) 119901

0)

times (119883119906 minus 119901

0) 119889120579]119883120593

120572

119889120585

= ∮

119861120588(1205850)

[119860

120572

119894(120585

0 119906

0 119883119906)

minus119860

120572

119894(120585

0 119906

0 119901

0)]119883120593

120572

119889120585

= ∮

119861120588(1205850)

[119860

120572

119894(120585

0 119906

0 119883119906)

minus119860

120572

119894(120585 119906 119883119906)]119883120593

120572

119889120585

+ ∮

119861120588(1205850)

119861

120572

(120585 119906 119883119906) 120593

120572

119889120585

(43)

It yields

∮

119861120588(1205850)

119860

120572

119894119901119895

120573

(120585

0 119906

0 119901

0) (119883119906 minus 119901

0)119883120593

120572

119889120585

= ∮

119861120588(1205850)

[int

1

0

(119860

120572

119894119901119895

120573

(120585

0 119906

0 119901

0)

minus119860

120572

119894119901119895

120573

(120585

0 119906

0 120579119883119906 + (1 minus 120579) 119901

0))

times (119883119906 minus 119901

0) 119889120579] 119889120585 sup

119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

+ ∮

119861120588(1205850)

[119860

120572

119894(120585

0 119906

0 119883119906)

minus119860

120572

119894(120585 119906

0+ 119901

0(120585 minus 120585

0) 119883119906)] sup

119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

+ ∮

119861120588(1205850)

[119860

120572

119894(120585 119906

0+ 119901

0(120585 minus 120585

0) 119883119906)

minus119860

120572

119894(120585 119906 119883119906)] sup

119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

+ ∮

119861120588(1205850)

119861

120572

(120585 119906 119883119906) 120593

120572

119889120585

= 119868

1015840

+ 119868119868

1015840

+ 119868119868119868

1015840

+ 119868119881

1015840

(44)

Using (22) Holderrsquos inequality the fact that 119905 rarr 120596

2

(119904 119905) isconcave and Jensenrsquos inequality we have

119868

1015840

le sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

∮

119861120588(1205850)

120596 (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

119889120585

le sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

[∮

119861120588(1205850)

120596

2

(

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

) 119889120585]

12

times[∮

119861120588(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

119889120585]

12

le 120596(

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

Φ (120585

0 120588 119901

0))Φ

12

(120585

0 120588 119901

0) sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

(45)

Similarly using (21) and the fact that 120578(119905119904) le 119905120578(119904) for 119905 ge 1we obtain

119868119868

1015840

le sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

120581 (sdot)radic120578 (120588

2

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

2

))

times ∮

119861120588(1205850)

(1 + |119883119906|) 119889120585

le sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

120581 (sdot)radic120578 (120588

2

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

)

times ∮

119861120588(1205850)

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

) 119889120585

le sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

[(∮

119861120588(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

119889120585)

+ 120581

2

(sdot) (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

120578 (120588

2

)

+ 120581 (sdot) (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

3

radic120578 (120588

2

) ]

le [Φ (120585

0 120588 119901

0)+ 2120581

2

(sdot) (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

3

radic120578 (120588

2

)]

times sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

(46)

8 Abstract and Applied Analysis

where we have used the fact that 120578(1205882) le

radic

120578(120588

2

) whichfollows from the nondecreasing property of the function 120578(119905)(1205784) and our assumption 120588 le 120588

1le 1

In the same way it follows that by using (21) (37) and(19)

119868119868119868

1015840

le sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

120581 (sdot)∮

119861120588(1205850)

radic120578 (|V|2) (1 + |119883119906|) 119889120585

le sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

[∮

119861120588(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

119889120585

+ 120581

2

(sdot)∮

119861120588(1205850)

120578 (|V|2) 119889120585

+ 120581 (sdot) (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)∮

119861120588(1205850)

radic120578 (|V|2) 119889120585]

le sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

[Φ (120585

0 120588 119901

0)

+ 2120588

minus2

∮

119861120588(1205850)

|V|2119889120585 + 1205814 (sdot) 120578 (1205882)

+ 120581

2

(sdot) (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

radic120578 (120588

2

) ]

le [(1 + 2119862

119901)Φ (120585

0 120588 119901

0)

+ 2120581

4

(sdot) (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

radic120578 (120588

2

)]

(47)

Using Holderrsquos inequality (19) and Youngrsquos inequality wehave

119868119881

1015840

le 119862∮

119861120588(1205850)

(1 + |119906|

119903minus1

+ |119883119906|

2(1minus1119903)

)

1003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119889120585

le 119862∮

119861120588(1205850)

|119883119906|

2(1minus1119903) 10038161003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119889120585

+ 119862∮

119861120588(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus 119906

0minus 119901

0(120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

119903minus11003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119889120585

+ 119862120588

2

[1 + (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

119903minus1

]

le 119862(∮

119861120588(1205850)

|119883119906|

2

119889120585)

(1minus1119903)

(∮

119861120588(1205850)

1003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119903

119889120585)

1119903

+ 119862(∮

119861120588(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus 119906

0minus 119901

0(120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

119903

119889120585)

(1minus1119903)

times (∮

119861120588(1205850)

1003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119903

119889120585)

1119903

+ 119862120588

2

[1 + (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

119903minus1

]

le 119862(∮

119861120588(1205850)

|119883119906|

2

119889120585)

(1minus1119903)

(∮

119861120588(1205850)

1003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119903

119889120585)

1119903

+ (∮

119861120588(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

119889120585)

(1199032)(1minus1119903)

times (∮

119861120588(1205850)

1003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119903

119889120585)

1119903

+ 119862120588

2

[1 + (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

119903minus1

]

le 119862119906

11986711988212(∮

119861120588(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

119889120585)

(1minus1119903)

times (∮

119861120588(1205850)

1003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119903

119889120585)

1119903

+ 119862120588 [1 + (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

119903minus1

]

le 119862119906

11986711988212(119861120588(1205850))

(∮

119861120588(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

119889120585)

+ 119862120588

2119903

+ 119862120588

2

[1 + (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

119903minus1

]

le 119862

2Φ(120585

0 120588 119901

0) + 119862120588

2

(1 +

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

119903minus1

le 119862

2Φ(120585

0 120588 119901

0) + 119862120581 (sdot) (1 +

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

radic120578 (120588

2

)

(48)

where we have used the assumption (1205784) and the fact that 119903 =2119876(119876 minus 2) = (2119899 + 4)2119899 le 3 and 119862

2= 119862119906

11986711988212(119861120588(1205850))

ge

1 Combining these estimates we obtain the conclusion with119862

1= (1 + 119862

2+ 2119862

119901) ge 1

Lemma 5 Assume that the conditions of Lemma 2 and thefollowing smallness conditions hold

120596 (

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

1003816

(119883119906)

1205850120588

1003816

1003816

1003816

1003816

1003816

Φ (120585

0 120588 (119883119906)

1205850120588))

+ Φ

12

(120585

0 120588 (119883119906)

1205850120588) le

120575

2

(49)

119862

3119870

2

1(

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

1205850120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

) le 120575

2 (50)

with 1198623= 8119862

2

1119862

5 together with

120588 le 120588

1(1 +

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

1 +

1003816

1003816

1003816

1003816

1003816

(119883119906)

1205850120588

1003816

1003816

1003816

1003816

1003816

) (51)

Then the following growth condition holds for 120591 isin [120574 1)

Φ (120585

0 120579120588) le 120579

2120591

Φ(120585

0 120588)

+ 119870

lowast

(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

)

(52)

where one abbreviates Φ(1205850 119903) = Φ(120585

0 119903 (119883119906)

1205850119903) and

119870

lowast

(119904 119905) = 119870(119904 119905) + (2 + 119904 + 119905)

2(119903minus1) with 119870(119904 119905) = (4120575

minus2

+

2

119876

119862

119888)119870

2

1(1 + 119904 1 + 119905)

Proof We define 119908 = [119906 minus 1199061205850120588

minus 119901

0(120585

1

minus 120585

1

0)]120590

minus1 where

120590 = 119862

1radic

Φ(120585

0 120588 119901

0) + 4120575

minus2119870

2

1(

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

) 120578 (120588

2)

(53)

Abstract and Applied Analysis 9

Then we have 119883119908 = 120590

minus1

(119883119906 minus 119901

0) Now we consider

119861

120588(120585

0) subsub Ω such that 120588 le 120588

1(|119906

0| |119901

0|) Applying Lemma 4

on 119861120588(120585

0) to 119906 we have for any 120593 isin 119862infin

0(119861

120588(120585

0)R119873)

∮

119861120588(1205850)

|119883119908|

2

119889120585 = 120590

minus2

Φ(120585

0 120588 119901

0) le

1

119862

2

1

le 1 (54)

∮

119861120588(1205850)

119860

120572

119894119901119895

120573

(120585

0 119906

0 119901

0)119883119908119883120593119889120585

le [Φ

12

(120585

0 120588 119901

0)

+120596 (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

Φ (120585

0 120588 119901

0)) +

120575

2

] sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

(55)

In consideration of the small condition (49) we see that(54) and (55) imply conditions (26) in Lemma 2 Also notethat (H1) and (H3) imply condition (25) So there exists an119860

120572

119894119901119895

120573

(120585

0 119906

0 119901

0)-harmonic function ℎ isin 11986711988212(119861

120588(120585

0)R119873)

such that

∮

119861120588(1205850)

|119883ℎ|

2

119889120585 le 1 120588

minus2

∮

119861120588(1205850)

|119908 minus ℎ|

2

119889120585 le 120576 (56)

Taking 1199060= 119906

12058502120579120588 120579 isin (0 14] and replacing 119901

0by 1199010+

120590(119883ℎ)

1205850 2120579120588 we use Lemma 3 to obtain

int

119861120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119883119906 minus 119901

0minus 120590(119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

2

119889120585

le 119862

119888[

1

(120579120588)

2int

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus 119906

12058502120579120588

minus (119901

0+ 120590(119883ℎ)

12058502120579120588)

times (120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

2

119889120585 + 119865]

(57)

where

119865 = 120596

119899(2120579120588)

119876

119870

1(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119901

0+ 120590(119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 ((2120579120588)

2

)

+ [int

1198612120579120588(1205850)

(1 + 119906

119903

+ |119883119906|

2

) 119889120585]

2(1minus1119903)

(58)

Using the fact that 119906 minus (1199010+ 120590(119883ℎ)

12058502120579120588)(120585

1

minus 120585

1

0) has mean

value 11990612058502120579120588

on the ball 1198612120579120588(120585

0) the definition of119908 and (19)

we have

1

(120579120588)

2∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus 119906

1205850 2120579120588minus (119901

0+ 120590(119883ℎ)

12058502120579120588) (120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

2

119889120585

le

4120590

2

(2120579120588)

2∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119908 minus ℎ

1205850 2120579120588

minus(119883ℎ)

12058502120579120588(120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

2

119889120585

le

4120590

2

(2120579120588)

2[∮

1198612120579120588(1205850)

|119908 minus ℎ|

2

119889120585

+ ∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

ℎ minus ℎ

1205850 2120579120588

minus(119883ℎ)

12058502120579120588(120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

2

119889120585]

le 4120590

2

[(2120579)

minus119876minus2

120576 + 119862

119901∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119883ℎ minus (119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

2

119889120585]

le 4120590

2

[(2120579)

minus119876minus2

120576 + 119862

2

119901(2120579120588)

2

∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119883

2

ℎ

1003816

1003816

1003816

1003816

1003816

2

119889120585]

le 4120590

2

[(2120579)

minus119876minus2

120576 + 119862

2

119901(2120579)

2

119862

0]

le 119862

4(120579

minus119876minus2

120576 + 120579

2

) [Φ (120585

0 120588 119901

0)

+ 4120575

minus2

119870

2

1(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

) 120578 (120588

2

)]

(59)

where 1198624= 119862

4(119876 120582 119871) ge 1 Note that in the second last

inequality we have used the fact that

∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119883

2

ℎ

1003816

1003816

1003816

1003816

1003816

119889120585 le sup119861120588(1205850)

1003816

1003816

1003816

1003816

1003816

119883

2

ℎ

1003816

1003816

1003816

1003816

1003816

le 119862

0120588

minus2

∮

119861120588(1205850)

|119883ℎ|

2

119889120585 le 119862

0120588

minus2

(60)

In consideration of the fact that 119903 = 2119876(119876 minus 2) gt 2 119876 ge 4

and the assumptions 120579 isin (0 14] and Φ le 1 it follows that

[∮

1198612120579120588(1205850)

(1 + 119906

119903

+ |119883119906|

2

) 119889120585]

2(1minus1119903)

le 119862[∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

119889120585]

2(1minus1119903)

+ 119862(∮

1198612120579120588(1205850)

|119883119906|

2

119889120585)

119903minus1

+ (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

4(1minus1119903)

)

10 Abstract and Applied Analysis

le 119862 [(2120579)

minus2119876(1minus1119903)

Φ(120585

0 120588 119901

0)

2(1minus1119903)

+(2120579)

minus119876(119903minus1)

Φ(120585

0 120588 119901

0)

119903minus1

]

+ (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

2(1minus1119903)

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

4(1minus1119903)

)

le 119862(2120579)

minus119876(119903minus1)

Φ(120585

0 120588 119901

0)

2(1minus1119903)

+ (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

2(1minus1119903)

)

2

(61)

Let 119875 = 1199010+120590(119883ℎ)

12058502120579120588with 119901

0= (119883119906)

12058502120579120588 Combining

these estimates (57)ndash(61) and considering the small condition(51) (it implies 120588 le 120588

1(|119906

12058502120579120588| |119875|) see (64) and (65)) we

deduce that

Φ(120585

0 120579120588) le

1003816

1003816

1003816

1003816

1003816

119861

120579120588(120585

0)

1003816

1003816

1003816

1003816

1003816

minus1

H119899int

1198612120579120588(1205850)

|119883119906 minus 119875|

2

119889120585

le 119862

119888

2

119876

(120579120588)

2∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus 119906

12058502120579120588

minus (119901

0+ 120590(119883ℎ)

12058502120579120588)

times (120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

2

119889120585

+ 2

119876

119862

119888119870

1(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119901

0+ 120590(119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

)

+ 119862

119888

(2120579120588)

2119876(1minus1119903)

(120579120588)

119876

times[∮

1198612120579120588(1205850)

(1 + 119906

119903

+ |119883119906|

2

) 119889120585]

2(1minus1119903)

le 2

119876

119862

4119862

119888(120579

minus119876minus2

120576 + 120579

2

)

times [Φ (120585

0 120588) + 4120575

minus2

119870

2

1

times (

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

) ]

+ 2

119876

119862

119888119870

1(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

+120590(119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

)

+ [2

119876

119862

119888(2120579)

2minus119876(119903minus1)

Φ(120585

0 120588)

2(1minus1119903)

+(1 +

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

2(1minus1119903)

)

2

] 120588

2

(62)

Wenow specify 120576 = 120579119876+4 120579 isin (0 14] such that 2119876+11198624119862

119888120579

2

le

120579

2120591 Note that the small condition (50) implies 12059021198625le 1 with

119862

5= max119862

0 119862

1198882

119876

(2120579)

minus(1198762+4)(119876minus2)

and it yields

2

119876

119862

119888(2120579)

2minus119876(119903minus1)

Φ(120585

0 120588)

2(1minus1119903)

le 1

(63)1003816

1003816

1003816

1003816

1003816

120590(119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

le 120590 sup1198612120579120588(1205850)

|119883ℎ|

le 120590radic119862

0(∮

119861120588(1205850)

|119883ℎ|

2

119889120585) le 120590radic119862

0le 1

(64)

where we have used the a priori estimate (28) for the A-harmonic function ℎ Furthermore using (19) and recallingthe definition of 120590 and 119862

1 we have

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

le

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588minus 119906

1205850120588

1003816

1003816

1003816

1003816

1003816

le

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+ (2120579)

minus1198762

times (∮

119861120588(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus (119883119906)

1205850120588(120585

1

minus 120585

1

0) minus 119906

1205850120588

1003816

1003816

1003816

1003816

1003816

2

119889120585)

12

le

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+ (2120579)

minus1198762

120588radic119862

119901Φ

12

(120585

0 120588)

le

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+

120590

radic

119862

119901

119862

1(2120579)

1198762

le

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+ 120590radic119862

5le

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+ 1

(65)

Combining these estimates with (62) we have

Φ(120585

0 120579120588) le 120579

2120591

Φ(120585

0 120588)

+ [4120575

minus2

119870

2

1(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

)

+ 2

119876

119862

119888119870

1(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

+120590(119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

)] 120578 (120588

2

)

+ [1 + (1 +

1003816

1003816

1003816

1003816

1003816

(119883119906)

1205850 2120579120588

1003816

1003816

1003816

1003816

1003816

2(1minus1119903)

)

2

] 120578 (120588

2

)

le 120579

2120591

Φ(120585

0 120588) + 119870 (

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

)

+ (2 +

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

)

2(119903minus1)

120578 (120588

2

)

le 120579

2120591

Φ(120585

0 120588) + 119870

lowast

(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

)

(66)

Then the proof of Lemma 5 is complete

Abstract and Applied Analysis 11

For 119879 gt 0 we find Φ0(119879) gt 0 (depending on 119876119873 120582 119871

120591 and 120596) such that

120596

2

(2119879 2Φ

0(119879)) + 2Φ

0(119879) le

1

2

120575

2

119862

1Φ

0(119879) le 120579

119876

(1 minus 120579

120591

)

2

119879

2

(67)

With Φ0(119879) from (67) we choose 120588

0(119879) isin (0 1] (depending

on 119876119873 120582 119871 120591 120596 120578 and 120581) such that

120588

0(119879) le 120588

1(1 + 2119879 1 + 2119879)

119862

3119870

2

1(2119879 2119879) 120578 (120588

0(119879)

2

) le 120575

2

119870

0(119879) 120578 (120588

0(119879)

2

) le (120579

2120574

minus 120579

2120591

)Φ

0(119879)

2 (1 + 119862

119901)119870

0(119879)119867 (120588

0(119879)

2

) le 120579

119876

(1 minus 120579

120574

)

2

(120579

2120574

minus 120579

2120591

) 119879

2

(68)

where1198700(119879) = 119870

lowast

(2119879 2119879)By the proof method of of Lemma 51 in [12] and

conditions (67)-(68) Lemma 6 can be proved As is wellknown it is sufficient to complete the proof ofTheorem 1 oncewe obtain Lemma 6

Lemma 6 Assume that for some 1198790gt 0 and 119861

120588(120585

0) subsub Ω one

has

(1) |1199061205850120588| + |(119883119906)

1205850120588| le 119879

0

(2) 120588 le 1205880(119879

0)

(3) Φ(1205850 120588) le Φ

0(119879

0)

Then the small conditions (49)ndash(51) are satisfied on the balls119861

120579119895120588(120585

0) for 119895 isin 119873 cup 0 Moreover the limit Λ

1205850=

lim119895rarrinfin

(119883119906)

1205850120579119895120588exists and the inequality

∮

119861120588(1205850)

1003816

1003816

1003816

1003816

1003816

119883119906 minus Λ

1205850

1003816

1003816

1003816

1003816

1003816

2

119889120585 le 119862

6((

119903

120588

)

2120591

Φ(120585

0 120588) + 119867 (119903

2

))

(69)

is valid for 0 lt 119903 le 120588 with a constant 1198626= 119862

6(119876 119873 120582 119871

120591 119886119899119889 119879

0)

Proof The proof is very similar to the proof of Lemma 51 in[12] We omit it here

Acknowledgments

The project was supported by the National Natural ScienceFoundation of China (no 11201081 and no 11126294) andby the Science and Technology Planning Project of JiangxiProvince China no GJJ13657

References

[1] J Wang and P Niu ldquoOptimal partial regularity for weaksolutions of nonlinear sub-elliptic systems in Carnot groupsrdquoNonlinear Analysis Theory Methods amp Applications vol 72 no11 pp 4162ndash4187 2010

[2] J Wang and D Liao ldquoOptimal partial regularity for sub-elliptic systems with sub-quadratic growth in Carnot groupsrdquoNonlinear Analysis Theory Methods amp Applications vol 75 no4 pp 2499ndash2519 2012

[3] E de Giorgi ldquoUn esempio di estremali discontinue per unproblema variazionale di tipo ellitticordquo Bollettino della UnioneMatematica Italiana vol 4 pp 135ndash137 1968

[4] M Giaquinta Multiple Integrals in the Calculus of Variationsand Nonlinear Elliptic Systems Princeton University PressPrinceton NJ USA 1983

[5] Y Chen and L Wu Second Order Elliptic Equations and EllipticSystems Science Press Beijing China 2003

[6] F Duzaar and K Steffen ldquoOptimal interior and boundary reg-ularity for almost minimizers to elliptic variational integralsrdquoJournal fur die Reine und Angewandte Mathematik vol 546 pp73ndash138 2002

[7] F Duzaar and J F Grotowski ldquoOptimal interior partial regu-larity for nonlinear elliptic systems the method of A-harmonicapproximationrdquo Manuscripta Mathematica vol 103 no 3 pp267ndash298 2000

[8] F Duzaar J F Grotowski and M Kronz ldquoRegularity ofalmost minimizers of quasi-convex variational integrals withsubquadratic growthrdquo Annali di Matematica Pura ed ApplicataIV vol 184 no 4 pp 421ndash448 2005

[9] F Duzaar and G Mingione ldquoThe p-harmonic approximationand the regularity of p-harmonic mapsrdquo Calculus of Variationsand Partial Differential Equations vol 20 no 3 pp 235ndash2562004

[10] F Duzaar and G Mingione ldquoRegularity for degenerate ellipticproblems via p-harmonic approximationrdquo Annales de lrsquoInstitutHenri Poincare Analyse Non Lineaire vol 21 no 5 pp 735ndash7662004

[11] S Chen andZ Tan ldquoThemethod ofA-harmonic approximationand optimal interior partial regularity for nonlinear ellipticsystems under the controllable growth conditionrdquo Journal ofMathematical Analysis and Applications vol 335 no 1 pp 20ndash42 2007

[12] F Duzaar and A Gastel ldquoNonlinear elliptic systems with Dinicontinuous coefficientsrdquo Archiv der Mathematik vol 78 no 1pp 58ndash73 2002

[13] F Duzaar A Gastel and G Mingione ldquoElliptic systemssingular sets and Dini continuityrdquo Communications in PartialDifferential Equations vol 29 no 7-8 pp 1215ndash1240 2004

[14] Y Qiu and Z Tan ldquoOptimal interior partial regularity fornonlinear elliptic systems with Dini continuous coefficientsrdquoActa Mathematica Scientia B vol 30 no 5 pp 1541ndash1554 2010

[15] Y Qiu ldquoOptimal partial regularity of second order nonlin-ear elliptic systems with Dini continuous coefficients for thesuperquadratic caserdquo Nonlinear Analysis Theory Methods ampApplications vol 75 no 8 pp 3574ndash3590 2012

[16] L Capogna ldquoRegularity of quasi-linear equations in theHeisen-berg grouprdquoCommunications on Pure andAppliedMathematicsvol 50 no 9 pp 867ndash889 1997

[17] L Capogna ldquoRegularity for quasilinear equations and 1-quasiconformal maps in Carnot groupsrdquo MathematischeAnnalen vol 313 no 2 pp 263ndash295 1999

[18] S Marchi ldquo1198621120572 local regularity for the solutions of the p-Laplacian on the Heisenberg group for 2 lt 119901 lt 1 +

radic

5rdquoZeitschrift fur Analysis und ihre Anwendungen vol 20 no 3 pp617ndash636 2001

12 Abstract and Applied Analysis

[19] S Marchi ldquo1198621120572 local regularity for the solutions of the p-Laplacian on the Heisenberg groupThe case 1+1radic5 lt 119901 le 2rdquoCommentationes Mathematicae Universitatis Carolinae vol 44no 1 pp 33ndash56 2003

[20] S Marchi ldquoL119901 regularity of the derivative in the secondcommutator direction for nonlinear elliptic equations on theHeisenberg grouprdquo Accademia Nazionale delle Scienze detta deiXL Rendiconti Serie V Memorie di Matematica e ApplicazioniParte I vol 26 pp 1ndash15 2002

[21] A Domokos ldquoDifferentiability of solutions for the non-degenerate p-Laplacian in the Heisenberg grouprdquo Journal ofDifferential Equations vol 204 no 2 pp 439ndash470 2004

[22] A Domokos On the regularity of p-harmonic functions inthe Heisenberg group [PhD thesis] University of PittsburghPittsburgh Pa USA 2004

[23] J J Manfredi and G Mingione ldquoRegularity results for quasilin-ear elliptic equations in the Heisenberg grouprdquo MathematischeAnnalen vol 339 no 3 pp 485ndash544 2007

[24] G Mingione A Zatorska-Goldstein and X Zhong ldquoGradientregularity for elliptic equations in the Heisenberg grouprdquoAdvances in Mathematics vol 222 no 1 pp 62ndash129 2009

[25] L Capogna and N Garofalo ldquoRegularity of minimizers of thecalculus of variations inCarnot groups via hypoellipticity of sys-tems ofHormander typerdquo Journal of the EuropeanMathematicalSociety vol 5 no 1 pp 1ndash40 2003

[26] E Shores ldquoHypoellipticity forlinear degenerate elliptic systemsin Carnot groups and applicationsrdquo httparxivorgabsmath0502569

[27] A Foglein ldquoPartial regularity results for subelliptic systemsin the Heisenberg grouprdquo Calculus of Variations and PartialDifferential Equations vol 32 no 1 pp 25ndash51 2008

[28] G Lu ldquoThe sharp Poincare inequality for free vector fields anendpoint resultrdquo Revista Matematica Iberoamericana vol 10no 2 pp 453ndash466 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 7: Research Article Partial Regularity for Nonlinear Subelliptic Systems …downloads.hindawi.com/journals/aaa/2013/950134.pdf · 2019-07-31 · Partial Regularity for Nonlinear Subelliptic

Abstract and Applied Analysis 7

Lemma 4 Let 119861120588(120585

0) subsub Ω with 120588 le 120588

1(|119906

0| |119901

0|) and 120593 isin

119862

infin

0(119861

120588(120585

0)R119873) satisfying |120593| le 120588

2 and sup119861120588(1205850)

|119883120593| le 1Then there exists a constant 119862

1ge 1 such that

∮

119861120588(1205850)

119860

120572

119894119901119895

120573

(120585

0 119906

0 119901

0) (119883119906 minus 119901

0)119883120593

120572

119889120585

le 119862

1[Φ (120585

0 120588 119901

0)

+ 120596 (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

Φ (120585

0 120588 119901

0))Φ

12

(120585

0 120588 119901

0)

+ 119870

1(

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)radic120578 (120588

2

)] sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

(42)

Proof Using the fact that int119861120588(1205850)

119860

120572

119894(120585

0 119906

0 119901

0)119883120593

120572

119889120585 = 0 andthe weak form (8) we deduce

∮

119861120588(1205850)

[int

1

0

119860

120572

119894119901119895

120573

(120585

0 119906

0 120579119883119906 + (1 minus 120579) 119901

0)

times (119883119906 minus 119901

0) 119889120579]119883120593

120572

119889120585

= ∮

119861120588(1205850)

[119860

120572

119894(120585

0 119906

0 119883119906)

minus119860

120572

119894(120585

0 119906

0 119901

0)]119883120593

120572

119889120585

= ∮

119861120588(1205850)

[119860

120572

119894(120585

0 119906

0 119883119906)

minus119860

120572

119894(120585 119906 119883119906)]119883120593

120572

119889120585

+ ∮

119861120588(1205850)

119861

120572

(120585 119906 119883119906) 120593

120572

119889120585

(43)

It yields

∮

119861120588(1205850)

119860

120572

119894119901119895

120573

(120585

0 119906

0 119901

0) (119883119906 minus 119901

0)119883120593

120572

119889120585

= ∮

119861120588(1205850)

[int

1

0

(119860

120572

119894119901119895

120573

(120585

0 119906

0 119901

0)

minus119860

120572

119894119901119895

120573

(120585

0 119906

0 120579119883119906 + (1 minus 120579) 119901

0))

times (119883119906 minus 119901

0) 119889120579] 119889120585 sup

119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

+ ∮

119861120588(1205850)

[119860

120572

119894(120585

0 119906

0 119883119906)

minus119860

120572

119894(120585 119906

0+ 119901

0(120585 minus 120585

0) 119883119906)] sup

119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

+ ∮

119861120588(1205850)

[119860

120572

119894(120585 119906

0+ 119901

0(120585 minus 120585

0) 119883119906)

minus119860

120572

119894(120585 119906 119883119906)] sup

119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

+ ∮

119861120588(1205850)

119861

120572

(120585 119906 119883119906) 120593

120572

119889120585

= 119868

1015840

+ 119868119868

1015840

+ 119868119868119868

1015840

+ 119868119881

1015840

(44)

Using (22) Holderrsquos inequality the fact that 119905 rarr 120596

2

(119904 119905) isconcave and Jensenrsquos inequality we have

119868

1015840

le sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

∮

119861120588(1205850)

120596 (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

119889120585

le sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

[∮

119861120588(1205850)

120596

2

(

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

) 119889120585]

12

times[∮

119861120588(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

119889120585]

12

le 120596(

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

Φ (120585

0 120588 119901

0))Φ

12

(120585

0 120588 119901

0) sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

(45)

Similarly using (21) and the fact that 120578(119905119904) le 119905120578(119904) for 119905 ge 1we obtain

119868119868

1015840

le sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

120581 (sdot)radic120578 (120588

2

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

2

))

times ∮

119861120588(1205850)

(1 + |119883119906|) 119889120585

le sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

120581 (sdot)radic120578 (120588

2

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

)

times ∮

119861120588(1205850)

(1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

) 119889120585

le sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

[(∮

119861120588(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

119889120585)

+ 120581

2

(sdot) (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

120578 (120588

2

)

+ 120581 (sdot) (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

3

radic120578 (120588

2

) ]

le [Φ (120585

0 120588 119901

0)+ 2120581

2

(sdot) (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

3

radic120578 (120588

2

)]

times sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

(46)

8 Abstract and Applied Analysis

where we have used the fact that 120578(1205882) le

radic

120578(120588

2

) whichfollows from the nondecreasing property of the function 120578(119905)(1205784) and our assumption 120588 le 120588

1le 1

In the same way it follows that by using (21) (37) and(19)

119868119868119868

1015840

le sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

120581 (sdot)∮

119861120588(1205850)

radic120578 (|V|2) (1 + |119883119906|) 119889120585

le sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

[∮

119861120588(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

119889120585

+ 120581

2

(sdot)∮

119861120588(1205850)

120578 (|V|2) 119889120585

+ 120581 (sdot) (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)∮

119861120588(1205850)

radic120578 (|V|2) 119889120585]

le sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

[Φ (120585

0 120588 119901

0)

+ 2120588

minus2

∮

119861120588(1205850)

|V|2119889120585 + 1205814 (sdot) 120578 (1205882)

+ 120581

2

(sdot) (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

radic120578 (120588

2

) ]

le [(1 + 2119862

119901)Φ (120585

0 120588 119901

0)

+ 2120581

4

(sdot) (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

radic120578 (120588

2

)]

(47)

Using Holderrsquos inequality (19) and Youngrsquos inequality wehave

119868119881

1015840

le 119862∮

119861120588(1205850)

(1 + |119906|

119903minus1

+ |119883119906|

2(1minus1119903)

)

1003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119889120585

le 119862∮

119861120588(1205850)

|119883119906|

2(1minus1119903) 10038161003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119889120585

+ 119862∮

119861120588(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus 119906

0minus 119901

0(120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

119903minus11003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119889120585

+ 119862120588

2

[1 + (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

119903minus1

]

le 119862(∮

119861120588(1205850)

|119883119906|

2

119889120585)

(1minus1119903)

(∮

119861120588(1205850)

1003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119903

119889120585)

1119903

+ 119862(∮

119861120588(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus 119906

0minus 119901

0(120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

119903

119889120585)

(1minus1119903)

times (∮

119861120588(1205850)

1003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119903

119889120585)

1119903

+ 119862120588

2

[1 + (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

119903minus1

]

le 119862(∮

119861120588(1205850)

|119883119906|

2

119889120585)

(1minus1119903)

(∮

119861120588(1205850)

1003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119903

119889120585)

1119903

+ (∮

119861120588(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

119889120585)

(1199032)(1minus1119903)

times (∮

119861120588(1205850)

1003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119903

119889120585)

1119903

+ 119862120588

2

[1 + (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

119903minus1

]

le 119862119906

11986711988212(∮

119861120588(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

119889120585)

(1minus1119903)

times (∮

119861120588(1205850)

1003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119903

119889120585)

1119903

+ 119862120588 [1 + (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

119903minus1

]

le 119862119906

11986711988212(119861120588(1205850))

(∮

119861120588(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

119889120585)

+ 119862120588

2119903

+ 119862120588

2

[1 + (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

119903minus1

]

le 119862

2Φ(120585

0 120588 119901

0) + 119862120588

2

(1 +

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

119903minus1

le 119862

2Φ(120585

0 120588 119901

0) + 119862120581 (sdot) (1 +

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

radic120578 (120588

2

)

(48)

where we have used the assumption (1205784) and the fact that 119903 =2119876(119876 minus 2) = (2119899 + 4)2119899 le 3 and 119862

2= 119862119906

11986711988212(119861120588(1205850))

ge

1 Combining these estimates we obtain the conclusion with119862

1= (1 + 119862

2+ 2119862

119901) ge 1

Lemma 5 Assume that the conditions of Lemma 2 and thefollowing smallness conditions hold

120596 (

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

1003816

(119883119906)

1205850120588

1003816

1003816

1003816

1003816

1003816

Φ (120585

0 120588 (119883119906)

1205850120588))

+ Φ

12

(120585

0 120588 (119883119906)

1205850120588) le

120575

2

(49)

119862

3119870

2

1(

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

1205850120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

) le 120575

2 (50)

with 1198623= 8119862

2

1119862

5 together with

120588 le 120588

1(1 +

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

1 +

1003816

1003816

1003816

1003816

1003816

(119883119906)

1205850120588

1003816

1003816

1003816

1003816

1003816

) (51)

Then the following growth condition holds for 120591 isin [120574 1)

Φ (120585

0 120579120588) le 120579

2120591

Φ(120585

0 120588)

+ 119870

lowast

(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

)

(52)

where one abbreviates Φ(1205850 119903) = Φ(120585

0 119903 (119883119906)

1205850119903) and

119870

lowast

(119904 119905) = 119870(119904 119905) + (2 + 119904 + 119905)

2(119903minus1) with 119870(119904 119905) = (4120575

minus2

+

2

119876

119862

119888)119870

2

1(1 + 119904 1 + 119905)

Proof We define 119908 = [119906 minus 1199061205850120588

minus 119901

0(120585

1

minus 120585

1

0)]120590

minus1 where

120590 = 119862

1radic

Φ(120585

0 120588 119901

0) + 4120575

minus2119870

2

1(

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

) 120578 (120588

2)

(53)

Abstract and Applied Analysis 9

Then we have 119883119908 = 120590

minus1

(119883119906 minus 119901

0) Now we consider

119861

120588(120585

0) subsub Ω such that 120588 le 120588

1(|119906

0| |119901

0|) Applying Lemma 4

on 119861120588(120585

0) to 119906 we have for any 120593 isin 119862infin

0(119861

120588(120585

0)R119873)

∮

119861120588(1205850)

|119883119908|

2

119889120585 = 120590

minus2

Φ(120585

0 120588 119901

0) le

1

119862

2

1

le 1 (54)

∮

119861120588(1205850)

119860

120572

119894119901119895

120573

(120585

0 119906

0 119901

0)119883119908119883120593119889120585

le [Φ

12

(120585

0 120588 119901

0)

+120596 (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

Φ (120585

0 120588 119901

0)) +

120575

2

] sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

(55)

In consideration of the small condition (49) we see that(54) and (55) imply conditions (26) in Lemma 2 Also notethat (H1) and (H3) imply condition (25) So there exists an119860

120572

119894119901119895

120573

(120585

0 119906

0 119901

0)-harmonic function ℎ isin 11986711988212(119861

120588(120585

0)R119873)

such that

∮

119861120588(1205850)

|119883ℎ|

2

119889120585 le 1 120588

minus2

∮

119861120588(1205850)

|119908 minus ℎ|

2

119889120585 le 120576 (56)

Taking 1199060= 119906

12058502120579120588 120579 isin (0 14] and replacing 119901

0by 1199010+

120590(119883ℎ)

1205850 2120579120588 we use Lemma 3 to obtain

int

119861120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119883119906 minus 119901

0minus 120590(119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

2

119889120585

le 119862

119888[

1

(120579120588)

2int

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus 119906

12058502120579120588

minus (119901

0+ 120590(119883ℎ)

12058502120579120588)

times (120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

2

119889120585 + 119865]

(57)

where

119865 = 120596

119899(2120579120588)

119876

119870

1(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119901

0+ 120590(119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 ((2120579120588)

2

)

+ [int

1198612120579120588(1205850)

(1 + 119906

119903

+ |119883119906|

2

) 119889120585]

2(1minus1119903)

(58)

Using the fact that 119906 minus (1199010+ 120590(119883ℎ)

12058502120579120588)(120585

1

minus 120585

1

0) has mean

value 11990612058502120579120588

on the ball 1198612120579120588(120585

0) the definition of119908 and (19)

we have

1

(120579120588)

2∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus 119906

1205850 2120579120588minus (119901

0+ 120590(119883ℎ)

12058502120579120588) (120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

2

119889120585

le

4120590

2

(2120579120588)

2∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119908 minus ℎ

1205850 2120579120588

minus(119883ℎ)

12058502120579120588(120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

2

119889120585

le

4120590

2

(2120579120588)

2[∮

1198612120579120588(1205850)

|119908 minus ℎ|

2

119889120585

+ ∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

ℎ minus ℎ

1205850 2120579120588

minus(119883ℎ)

12058502120579120588(120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

2

119889120585]

le 4120590

2

[(2120579)

minus119876minus2

120576 + 119862

119901∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119883ℎ minus (119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

2

119889120585]

le 4120590

2

[(2120579)

minus119876minus2

120576 + 119862

2

119901(2120579120588)

2

∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119883

2

ℎ

1003816

1003816

1003816

1003816

1003816

2

119889120585]

le 4120590

2

[(2120579)

minus119876minus2

120576 + 119862

2

119901(2120579)

2

119862

0]

le 119862

4(120579

minus119876minus2

120576 + 120579

2

) [Φ (120585

0 120588 119901

0)

+ 4120575

minus2

119870

2

1(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

) 120578 (120588

2

)]

(59)

where 1198624= 119862

4(119876 120582 119871) ge 1 Note that in the second last

inequality we have used the fact that

∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119883

2

ℎ

1003816

1003816

1003816

1003816

1003816

119889120585 le sup119861120588(1205850)

1003816

1003816

1003816

1003816

1003816

119883

2

ℎ

1003816

1003816

1003816

1003816

1003816

le 119862

0120588

minus2

∮

119861120588(1205850)

|119883ℎ|

2

119889120585 le 119862

0120588

minus2

(60)

In consideration of the fact that 119903 = 2119876(119876 minus 2) gt 2 119876 ge 4

and the assumptions 120579 isin (0 14] and Φ le 1 it follows that

[∮

1198612120579120588(1205850)

(1 + 119906

119903

+ |119883119906|

2

) 119889120585]

2(1minus1119903)

le 119862[∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

119889120585]

2(1minus1119903)

+ 119862(∮

1198612120579120588(1205850)

|119883119906|

2

119889120585)

119903minus1

+ (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

4(1minus1119903)

)

10 Abstract and Applied Analysis

le 119862 [(2120579)

minus2119876(1minus1119903)

Φ(120585

0 120588 119901

0)

2(1minus1119903)

+(2120579)

minus119876(119903minus1)

Φ(120585

0 120588 119901

0)

119903minus1

]

+ (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

2(1minus1119903)

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

4(1minus1119903)

)

le 119862(2120579)

minus119876(119903minus1)

Φ(120585

0 120588 119901

0)

2(1minus1119903)

+ (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

2(1minus1119903)

)

2

(61)

Let 119875 = 1199010+120590(119883ℎ)

12058502120579120588with 119901

0= (119883119906)

12058502120579120588 Combining

these estimates (57)ndash(61) and considering the small condition(51) (it implies 120588 le 120588

1(|119906

12058502120579120588| |119875|) see (64) and (65)) we

deduce that

Φ(120585

0 120579120588) le

1003816

1003816

1003816

1003816

1003816

119861

120579120588(120585

0)

1003816

1003816

1003816

1003816

1003816

minus1

H119899int

1198612120579120588(1205850)

|119883119906 minus 119875|

2

119889120585

le 119862

119888

2

119876

(120579120588)

2∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus 119906

12058502120579120588

minus (119901

0+ 120590(119883ℎ)

12058502120579120588)

times (120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

2

119889120585

+ 2

119876

119862

119888119870

1(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119901

0+ 120590(119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

)

+ 119862

119888

(2120579120588)

2119876(1minus1119903)

(120579120588)

119876

times[∮

1198612120579120588(1205850)

(1 + 119906

119903

+ |119883119906|

2

) 119889120585]

2(1minus1119903)

le 2

119876

119862

4119862

119888(120579

minus119876minus2

120576 + 120579

2

)

times [Φ (120585

0 120588) + 4120575

minus2

119870

2

1

times (

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

) ]

+ 2

119876

119862

119888119870

1(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

+120590(119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

)

+ [2

119876

119862

119888(2120579)

2minus119876(119903minus1)

Φ(120585

0 120588)

2(1minus1119903)

+(1 +

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

2(1minus1119903)

)

2

] 120588

2

(62)

Wenow specify 120576 = 120579119876+4 120579 isin (0 14] such that 2119876+11198624119862

119888120579

2

le

120579

2120591 Note that the small condition (50) implies 12059021198625le 1 with

119862

5= max119862

0 119862

1198882

119876

(2120579)

minus(1198762+4)(119876minus2)

and it yields

2

119876

119862

119888(2120579)

2minus119876(119903minus1)

Φ(120585

0 120588)

2(1minus1119903)

le 1

(63)1003816

1003816

1003816

1003816

1003816

120590(119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

le 120590 sup1198612120579120588(1205850)

|119883ℎ|

le 120590radic119862

0(∮

119861120588(1205850)

|119883ℎ|

2

119889120585) le 120590radic119862

0le 1

(64)

where we have used the a priori estimate (28) for the A-harmonic function ℎ Furthermore using (19) and recallingthe definition of 120590 and 119862

1 we have

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

le

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588minus 119906

1205850120588

1003816

1003816

1003816

1003816

1003816

le

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+ (2120579)

minus1198762

times (∮

119861120588(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus (119883119906)

1205850120588(120585

1

minus 120585

1

0) minus 119906

1205850120588

1003816

1003816

1003816

1003816

1003816

2

119889120585)

12

le

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+ (2120579)

minus1198762

120588radic119862

119901Φ

12

(120585

0 120588)

le

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+

120590

radic

119862

119901

119862

1(2120579)

1198762

le

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+ 120590radic119862

5le

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+ 1

(65)

Combining these estimates with (62) we have

Φ(120585

0 120579120588) le 120579

2120591

Φ(120585

0 120588)

+ [4120575

minus2

119870

2

1(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

)

+ 2

119876

119862

119888119870

1(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

+120590(119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

)] 120578 (120588

2

)

+ [1 + (1 +

1003816

1003816

1003816

1003816

1003816

(119883119906)

1205850 2120579120588

1003816

1003816

1003816

1003816

1003816

2(1minus1119903)

)

2

] 120578 (120588

2

)

le 120579

2120591

Φ(120585

0 120588) + 119870 (

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

)

+ (2 +

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

)

2(119903minus1)

120578 (120588

2

)

le 120579

2120591

Φ(120585

0 120588) + 119870

lowast

(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

)

(66)

Then the proof of Lemma 5 is complete

Abstract and Applied Analysis 11

For 119879 gt 0 we find Φ0(119879) gt 0 (depending on 119876119873 120582 119871

120591 and 120596) such that

120596

2

(2119879 2Φ

0(119879)) + 2Φ

0(119879) le

1

2

120575

2

119862

1Φ

0(119879) le 120579

119876

(1 minus 120579

120591

)

2

119879

2

(67)

With Φ0(119879) from (67) we choose 120588

0(119879) isin (0 1] (depending

on 119876119873 120582 119871 120591 120596 120578 and 120581) such that

120588

0(119879) le 120588

1(1 + 2119879 1 + 2119879)

119862

3119870

2

1(2119879 2119879) 120578 (120588

0(119879)

2

) le 120575

2

119870

0(119879) 120578 (120588

0(119879)

2

) le (120579

2120574

minus 120579

2120591

)Φ

0(119879)

2 (1 + 119862

119901)119870

0(119879)119867 (120588

0(119879)

2

) le 120579

119876

(1 minus 120579

120574

)

2

(120579

2120574

minus 120579

2120591

) 119879

2

(68)

where1198700(119879) = 119870

lowast

(2119879 2119879)By the proof method of of Lemma 51 in [12] and

conditions (67)-(68) Lemma 6 can be proved As is wellknown it is sufficient to complete the proof ofTheorem 1 oncewe obtain Lemma 6

Lemma 6 Assume that for some 1198790gt 0 and 119861

120588(120585

0) subsub Ω one

has

(1) |1199061205850120588| + |(119883119906)

1205850120588| le 119879

0

(2) 120588 le 1205880(119879

0)

(3) Φ(1205850 120588) le Φ

0(119879

0)

Then the small conditions (49)ndash(51) are satisfied on the balls119861

120579119895120588(120585

0) for 119895 isin 119873 cup 0 Moreover the limit Λ

1205850=

lim119895rarrinfin

(119883119906)

1205850120579119895120588exists and the inequality

∮

119861120588(1205850)

1003816

1003816

1003816

1003816

1003816

119883119906 minus Λ

1205850

1003816

1003816

1003816

1003816

1003816

2

119889120585 le 119862

6((

119903

120588

)

2120591

Φ(120585

0 120588) + 119867 (119903

2

))

(69)

is valid for 0 lt 119903 le 120588 with a constant 1198626= 119862

6(119876 119873 120582 119871

120591 119886119899119889 119879

0)

Proof The proof is very similar to the proof of Lemma 51 in[12] We omit it here

Acknowledgments

The project was supported by the National Natural ScienceFoundation of China (no 11201081 and no 11126294) andby the Science and Technology Planning Project of JiangxiProvince China no GJJ13657

References

[1] J Wang and P Niu ldquoOptimal partial regularity for weaksolutions of nonlinear sub-elliptic systems in Carnot groupsrdquoNonlinear Analysis Theory Methods amp Applications vol 72 no11 pp 4162ndash4187 2010

[2] J Wang and D Liao ldquoOptimal partial regularity for sub-elliptic systems with sub-quadratic growth in Carnot groupsrdquoNonlinear Analysis Theory Methods amp Applications vol 75 no4 pp 2499ndash2519 2012

[3] E de Giorgi ldquoUn esempio di estremali discontinue per unproblema variazionale di tipo ellitticordquo Bollettino della UnioneMatematica Italiana vol 4 pp 135ndash137 1968

[4] M Giaquinta Multiple Integrals in the Calculus of Variationsand Nonlinear Elliptic Systems Princeton University PressPrinceton NJ USA 1983

[5] Y Chen and L Wu Second Order Elliptic Equations and EllipticSystems Science Press Beijing China 2003

[6] F Duzaar and K Steffen ldquoOptimal interior and boundary reg-ularity for almost minimizers to elliptic variational integralsrdquoJournal fur die Reine und Angewandte Mathematik vol 546 pp73ndash138 2002

[7] F Duzaar and J F Grotowski ldquoOptimal interior partial regu-larity for nonlinear elliptic systems the method of A-harmonicapproximationrdquo Manuscripta Mathematica vol 103 no 3 pp267ndash298 2000

[8] F Duzaar J F Grotowski and M Kronz ldquoRegularity ofalmost minimizers of quasi-convex variational integrals withsubquadratic growthrdquo Annali di Matematica Pura ed ApplicataIV vol 184 no 4 pp 421ndash448 2005

[9] F Duzaar and G Mingione ldquoThe p-harmonic approximationand the regularity of p-harmonic mapsrdquo Calculus of Variationsand Partial Differential Equations vol 20 no 3 pp 235ndash2562004

[10] F Duzaar and G Mingione ldquoRegularity for degenerate ellipticproblems via p-harmonic approximationrdquo Annales de lrsquoInstitutHenri Poincare Analyse Non Lineaire vol 21 no 5 pp 735ndash7662004

[11] S Chen andZ Tan ldquoThemethod ofA-harmonic approximationand optimal interior partial regularity for nonlinear ellipticsystems under the controllable growth conditionrdquo Journal ofMathematical Analysis and Applications vol 335 no 1 pp 20ndash42 2007

[12] F Duzaar and A Gastel ldquoNonlinear elliptic systems with Dinicontinuous coefficientsrdquo Archiv der Mathematik vol 78 no 1pp 58ndash73 2002

[13] F Duzaar A Gastel and G Mingione ldquoElliptic systemssingular sets and Dini continuityrdquo Communications in PartialDifferential Equations vol 29 no 7-8 pp 1215ndash1240 2004

[14] Y Qiu and Z Tan ldquoOptimal interior partial regularity fornonlinear elliptic systems with Dini continuous coefficientsrdquoActa Mathematica Scientia B vol 30 no 5 pp 1541ndash1554 2010

[15] Y Qiu ldquoOptimal partial regularity of second order nonlin-ear elliptic systems with Dini continuous coefficients for thesuperquadratic caserdquo Nonlinear Analysis Theory Methods ampApplications vol 75 no 8 pp 3574ndash3590 2012

[16] L Capogna ldquoRegularity of quasi-linear equations in theHeisen-berg grouprdquoCommunications on Pure andAppliedMathematicsvol 50 no 9 pp 867ndash889 1997

[17] L Capogna ldquoRegularity for quasilinear equations and 1-quasiconformal maps in Carnot groupsrdquo MathematischeAnnalen vol 313 no 2 pp 263ndash295 1999

[18] S Marchi ldquo1198621120572 local regularity for the solutions of the p-Laplacian on the Heisenberg group for 2 lt 119901 lt 1 +

radic

5rdquoZeitschrift fur Analysis und ihre Anwendungen vol 20 no 3 pp617ndash636 2001

12 Abstract and Applied Analysis

[19] S Marchi ldquo1198621120572 local regularity for the solutions of the p-Laplacian on the Heisenberg groupThe case 1+1radic5 lt 119901 le 2rdquoCommentationes Mathematicae Universitatis Carolinae vol 44no 1 pp 33ndash56 2003

[20] S Marchi ldquoL119901 regularity of the derivative in the secondcommutator direction for nonlinear elliptic equations on theHeisenberg grouprdquo Accademia Nazionale delle Scienze detta deiXL Rendiconti Serie V Memorie di Matematica e ApplicazioniParte I vol 26 pp 1ndash15 2002

[21] A Domokos ldquoDifferentiability of solutions for the non-degenerate p-Laplacian in the Heisenberg grouprdquo Journal ofDifferential Equations vol 204 no 2 pp 439ndash470 2004

[22] A Domokos On the regularity of p-harmonic functions inthe Heisenberg group [PhD thesis] University of PittsburghPittsburgh Pa USA 2004

[23] J J Manfredi and G Mingione ldquoRegularity results for quasilin-ear elliptic equations in the Heisenberg grouprdquo MathematischeAnnalen vol 339 no 3 pp 485ndash544 2007

[24] G Mingione A Zatorska-Goldstein and X Zhong ldquoGradientregularity for elliptic equations in the Heisenberg grouprdquoAdvances in Mathematics vol 222 no 1 pp 62ndash129 2009

[25] L Capogna and N Garofalo ldquoRegularity of minimizers of thecalculus of variations inCarnot groups via hypoellipticity of sys-tems ofHormander typerdquo Journal of the EuropeanMathematicalSociety vol 5 no 1 pp 1ndash40 2003

[26] E Shores ldquoHypoellipticity forlinear degenerate elliptic systemsin Carnot groups and applicationsrdquo httparxivorgabsmath0502569

[27] A Foglein ldquoPartial regularity results for subelliptic systemsin the Heisenberg grouprdquo Calculus of Variations and PartialDifferential Equations vol 32 no 1 pp 25ndash51 2008

[28] G Lu ldquoThe sharp Poincare inequality for free vector fields anendpoint resultrdquo Revista Matematica Iberoamericana vol 10no 2 pp 453ndash466 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 8: Research Article Partial Regularity for Nonlinear Subelliptic Systems …downloads.hindawi.com/journals/aaa/2013/950134.pdf · 2019-07-31 · Partial Regularity for Nonlinear Subelliptic

8 Abstract and Applied Analysis

where we have used the fact that 120578(1205882) le

radic

120578(120588

2

) whichfollows from the nondecreasing property of the function 120578(119905)(1205784) and our assumption 120588 le 120588

1le 1

In the same way it follows that by using (21) (37) and(19)

119868119868119868

1015840

le sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

120581 (sdot)∮

119861120588(1205850)

radic120578 (|V|2) (1 + |119883119906|) 119889120585

le sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

[∮

119861120588(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

119889120585

+ 120581

2

(sdot)∮

119861120588(1205850)

120578 (|V|2) 119889120585

+ 120581 (sdot) (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)∮

119861120588(1205850)

radic120578 (|V|2) 119889120585]

le sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

[Φ (120585

0 120588 119901

0)

+ 2120588

minus2

∮

119861120588(1205850)

|V|2119889120585 + 1205814 (sdot) 120578 (1205882)

+ 120581

2

(sdot) (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

radic120578 (120588

2

) ]

le [(1 + 2119862

119901)Φ (120585

0 120588 119901

0)

+ 2120581

4

(sdot) (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

radic120578 (120588

2

)]

(47)

Using Holderrsquos inequality (19) and Youngrsquos inequality wehave

119868119881

1015840

le 119862∮

119861120588(1205850)

(1 + |119906|

119903minus1

+ |119883119906|

2(1minus1119903)

)

1003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119889120585

le 119862∮

119861120588(1205850)

|119883119906|

2(1minus1119903) 10038161003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119889120585

+ 119862∮

119861120588(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus 119906

0minus 119901

0(120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

119903minus11003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119889120585

+ 119862120588

2

[1 + (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

119903minus1

]

le 119862(∮

119861120588(1205850)

|119883119906|

2

119889120585)

(1minus1119903)

(∮

119861120588(1205850)

1003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119903

119889120585)

1119903

+ 119862(∮

119861120588(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus 119906

0minus 119901

0(120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

119903

119889120585)

(1minus1119903)

times (∮

119861120588(1205850)

1003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119903

119889120585)

1119903

+ 119862120588

2

[1 + (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

119903minus1

]

le 119862(∮

119861120588(1205850)

|119883119906|

2

119889120585)

(1minus1119903)

(∮

119861120588(1205850)

1003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119903

119889120585)

1119903

+ (∮

119861120588(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

119889120585)

(1199032)(1minus1119903)

times (∮

119861120588(1205850)

1003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119903

119889120585)

1119903

+ 119862120588

2

[1 + (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

119903minus1

]

le 119862119906

11986711988212(∮

119861120588(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

119889120585)

(1minus1119903)

times (∮

119861120588(1205850)

1003816

1003816

1003816

1003816

120593

1003816

1003816

1003816

1003816

119903

119889120585)

1119903

+ 119862120588 [1 + (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

119903minus1

]

le 119862119906

11986711988212(119861120588(1205850))

(∮

119861120588(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

119889120585)

+ 119862120588

2119903

+ 119862120588

2

[1 + (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

119903minus1

]

le 119862

2Φ(120585

0 120588 119901

0) + 119862120588

2

(1 +

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

119903minus1

le 119862

2Φ(120585

0 120588 119901

0) + 119862120581 (sdot) (1 +

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

)

2

radic120578 (120588

2

)

(48)

where we have used the assumption (1205784) and the fact that 119903 =2119876(119876 minus 2) = (2119899 + 4)2119899 le 3 and 119862

2= 119862119906

11986711988212(119861120588(1205850))

ge

1 Combining these estimates we obtain the conclusion with119862

1= (1 + 119862

2+ 2119862

119901) ge 1

Lemma 5 Assume that the conditions of Lemma 2 and thefollowing smallness conditions hold

120596 (

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

1003816

(119883119906)

1205850120588

1003816

1003816

1003816

1003816

1003816

Φ (120585

0 120588 (119883119906)

1205850120588))

+ Φ

12

(120585

0 120588 (119883119906)

1205850120588) le

120575

2

(49)

119862

3119870

2

1(

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

1205850120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

) le 120575

2 (50)

with 1198623= 8119862

2

1119862

5 together with

120588 le 120588

1(1 +

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

1 +

1003816

1003816

1003816

1003816

1003816

(119883119906)

1205850120588

1003816

1003816

1003816

1003816

1003816

) (51)

Then the following growth condition holds for 120591 isin [120574 1)

Φ (120585

0 120579120588) le 120579

2120591

Φ(120585

0 120588)

+ 119870

lowast

(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

)

(52)

where one abbreviates Φ(1205850 119903) = Φ(120585

0 119903 (119883119906)

1205850119903) and

119870

lowast

(119904 119905) = 119870(119904 119905) + (2 + 119904 + 119905)

2(119903minus1) with 119870(119904 119905) = (4120575

minus2

+

2

119876

119862

119888)119870

2

1(1 + 119904 1 + 119905)

Proof We define 119908 = [119906 minus 1199061205850120588

minus 119901

0(120585

1

minus 120585

1

0)]120590

minus1 where

120590 = 119862

1radic

Φ(120585

0 120588 119901

0) + 4120575

minus2119870

2

1(

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

) 120578 (120588

2)

(53)

Abstract and Applied Analysis 9

Then we have 119883119908 = 120590

minus1

(119883119906 minus 119901

0) Now we consider

119861

120588(120585

0) subsub Ω such that 120588 le 120588

1(|119906

0| |119901

0|) Applying Lemma 4

on 119861120588(120585

0) to 119906 we have for any 120593 isin 119862infin

0(119861

120588(120585

0)R119873)

∮

119861120588(1205850)

|119883119908|

2

119889120585 = 120590

minus2

Φ(120585

0 120588 119901

0) le

1

119862

2

1

le 1 (54)

∮

119861120588(1205850)

119860

120572

119894119901119895

120573

(120585

0 119906

0 119901

0)119883119908119883120593119889120585

le [Φ

12

(120585

0 120588 119901

0)

+120596 (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

Φ (120585

0 120588 119901

0)) +

120575

2

] sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

(55)

In consideration of the small condition (49) we see that(54) and (55) imply conditions (26) in Lemma 2 Also notethat (H1) and (H3) imply condition (25) So there exists an119860

120572

119894119901119895

120573

(120585

0 119906

0 119901

0)-harmonic function ℎ isin 11986711988212(119861

120588(120585

0)R119873)

such that

∮

119861120588(1205850)

|119883ℎ|

2

119889120585 le 1 120588

minus2

∮

119861120588(1205850)

|119908 minus ℎ|

2

119889120585 le 120576 (56)

Taking 1199060= 119906

12058502120579120588 120579 isin (0 14] and replacing 119901

0by 1199010+

120590(119883ℎ)

1205850 2120579120588 we use Lemma 3 to obtain

int

119861120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119883119906 minus 119901

0minus 120590(119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

2

119889120585

le 119862

119888[

1

(120579120588)

2int

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus 119906

12058502120579120588

minus (119901

0+ 120590(119883ℎ)

12058502120579120588)

times (120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

2

119889120585 + 119865]

(57)

where

119865 = 120596

119899(2120579120588)

119876

119870

1(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119901

0+ 120590(119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 ((2120579120588)

2

)

+ [int

1198612120579120588(1205850)

(1 + 119906

119903

+ |119883119906|

2

) 119889120585]

2(1minus1119903)

(58)

Using the fact that 119906 minus (1199010+ 120590(119883ℎ)

12058502120579120588)(120585

1

minus 120585

1

0) has mean

value 11990612058502120579120588

on the ball 1198612120579120588(120585

0) the definition of119908 and (19)

we have

1

(120579120588)

2∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus 119906

1205850 2120579120588minus (119901

0+ 120590(119883ℎ)

12058502120579120588) (120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

2

119889120585

le

4120590

2

(2120579120588)

2∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119908 minus ℎ

1205850 2120579120588

minus(119883ℎ)

12058502120579120588(120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

2

119889120585

le

4120590

2

(2120579120588)

2[∮

1198612120579120588(1205850)

|119908 minus ℎ|

2

119889120585

+ ∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

ℎ minus ℎ

1205850 2120579120588

minus(119883ℎ)

12058502120579120588(120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

2

119889120585]

le 4120590

2

[(2120579)

minus119876minus2

120576 + 119862

119901∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119883ℎ minus (119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

2

119889120585]

le 4120590

2

[(2120579)

minus119876minus2

120576 + 119862

2

119901(2120579120588)

2

∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119883

2

ℎ

1003816

1003816

1003816

1003816

1003816

2

119889120585]

le 4120590

2

[(2120579)

minus119876minus2

120576 + 119862

2

119901(2120579)

2

119862

0]

le 119862

4(120579

minus119876minus2

120576 + 120579

2

) [Φ (120585

0 120588 119901

0)

+ 4120575

minus2

119870

2

1(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

) 120578 (120588

2

)]

(59)

where 1198624= 119862

4(119876 120582 119871) ge 1 Note that in the second last

inequality we have used the fact that

∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119883

2

ℎ

1003816

1003816

1003816

1003816

1003816

119889120585 le sup119861120588(1205850)

1003816

1003816

1003816

1003816

1003816

119883

2

ℎ

1003816

1003816

1003816

1003816

1003816

le 119862

0120588

minus2

∮

119861120588(1205850)

|119883ℎ|

2

119889120585 le 119862

0120588

minus2

(60)

In consideration of the fact that 119903 = 2119876(119876 minus 2) gt 2 119876 ge 4

and the assumptions 120579 isin (0 14] and Φ le 1 it follows that

[∮

1198612120579120588(1205850)

(1 + 119906

119903

+ |119883119906|

2

) 119889120585]

2(1minus1119903)

le 119862[∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

119889120585]

2(1minus1119903)

+ 119862(∮

1198612120579120588(1205850)

|119883119906|

2

119889120585)

119903minus1

+ (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

4(1minus1119903)

)

10 Abstract and Applied Analysis

le 119862 [(2120579)

minus2119876(1minus1119903)

Φ(120585

0 120588 119901

0)

2(1minus1119903)

+(2120579)

minus119876(119903minus1)

Φ(120585

0 120588 119901

0)

119903minus1

]

+ (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

2(1minus1119903)

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

4(1minus1119903)

)

le 119862(2120579)

minus119876(119903minus1)

Φ(120585

0 120588 119901

0)

2(1minus1119903)

+ (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

2(1minus1119903)

)

2

(61)

Let 119875 = 1199010+120590(119883ℎ)

12058502120579120588with 119901

0= (119883119906)

12058502120579120588 Combining

these estimates (57)ndash(61) and considering the small condition(51) (it implies 120588 le 120588

1(|119906

12058502120579120588| |119875|) see (64) and (65)) we

deduce that

Φ(120585

0 120579120588) le

1003816

1003816

1003816

1003816

1003816

119861

120579120588(120585

0)

1003816

1003816

1003816

1003816

1003816

minus1

H119899int

1198612120579120588(1205850)

|119883119906 minus 119875|

2

119889120585

le 119862

119888

2

119876

(120579120588)

2∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus 119906

12058502120579120588

minus (119901

0+ 120590(119883ℎ)

12058502120579120588)

times (120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

2

119889120585

+ 2

119876

119862

119888119870

1(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119901

0+ 120590(119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

)

+ 119862

119888

(2120579120588)

2119876(1minus1119903)

(120579120588)

119876

times[∮

1198612120579120588(1205850)

(1 + 119906

119903

+ |119883119906|

2

) 119889120585]

2(1minus1119903)

le 2

119876

119862

4119862

119888(120579

minus119876minus2

120576 + 120579

2

)

times [Φ (120585

0 120588) + 4120575

minus2

119870

2

1

times (

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

) ]

+ 2

119876

119862

119888119870

1(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

+120590(119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

)

+ [2

119876

119862

119888(2120579)

2minus119876(119903minus1)

Φ(120585

0 120588)

2(1minus1119903)

+(1 +

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

2(1minus1119903)

)

2

] 120588

2

(62)

Wenow specify 120576 = 120579119876+4 120579 isin (0 14] such that 2119876+11198624119862

119888120579

2

le

120579

2120591 Note that the small condition (50) implies 12059021198625le 1 with

119862

5= max119862

0 119862

1198882

119876

(2120579)

minus(1198762+4)(119876minus2)

and it yields

2

119876

119862

119888(2120579)

2minus119876(119903minus1)

Φ(120585

0 120588)

2(1minus1119903)

le 1

(63)1003816

1003816

1003816

1003816

1003816

120590(119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

le 120590 sup1198612120579120588(1205850)

|119883ℎ|

le 120590radic119862

0(∮

119861120588(1205850)

|119883ℎ|

2

119889120585) le 120590radic119862

0le 1

(64)

where we have used the a priori estimate (28) for the A-harmonic function ℎ Furthermore using (19) and recallingthe definition of 120590 and 119862

1 we have

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

le

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588minus 119906

1205850120588

1003816

1003816

1003816

1003816

1003816

le

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+ (2120579)

minus1198762

times (∮

119861120588(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus (119883119906)

1205850120588(120585

1

minus 120585

1

0) minus 119906

1205850120588

1003816

1003816

1003816

1003816

1003816

2

119889120585)

12

le

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+ (2120579)

minus1198762

120588radic119862

119901Φ

12

(120585

0 120588)

le

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+

120590

radic

119862

119901

119862

1(2120579)

1198762

le

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+ 120590radic119862

5le

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+ 1

(65)

Combining these estimates with (62) we have

Φ(120585

0 120579120588) le 120579

2120591

Φ(120585

0 120588)

+ [4120575

minus2

119870

2

1(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

)

+ 2

119876

119862

119888119870

1(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

+120590(119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

)] 120578 (120588

2

)

+ [1 + (1 +

1003816

1003816

1003816

1003816

1003816

(119883119906)

1205850 2120579120588

1003816

1003816

1003816

1003816

1003816

2(1minus1119903)

)

2

] 120578 (120588

2

)

le 120579

2120591

Φ(120585

0 120588) + 119870 (

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

)

+ (2 +

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

)

2(119903minus1)

120578 (120588

2

)

le 120579

2120591

Φ(120585

0 120588) + 119870

lowast

(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

)

(66)

Then the proof of Lemma 5 is complete

Abstract and Applied Analysis 11

For 119879 gt 0 we find Φ0(119879) gt 0 (depending on 119876119873 120582 119871

120591 and 120596) such that

120596

2

(2119879 2Φ

0(119879)) + 2Φ

0(119879) le

1

2

120575

2

119862

1Φ

0(119879) le 120579

119876

(1 minus 120579

120591

)

2

119879

2

(67)

With Φ0(119879) from (67) we choose 120588

0(119879) isin (0 1] (depending

on 119876119873 120582 119871 120591 120596 120578 and 120581) such that

120588

0(119879) le 120588

1(1 + 2119879 1 + 2119879)

119862

3119870

2

1(2119879 2119879) 120578 (120588

0(119879)

2

) le 120575

2

119870

0(119879) 120578 (120588

0(119879)

2

) le (120579

2120574

minus 120579

2120591

)Φ

0(119879)

2 (1 + 119862

119901)119870

0(119879)119867 (120588

0(119879)

2

) le 120579

119876

(1 minus 120579

120574

)

2

(120579

2120574

minus 120579

2120591

) 119879

2

(68)

where1198700(119879) = 119870

lowast

(2119879 2119879)By the proof method of of Lemma 51 in [12] and

conditions (67)-(68) Lemma 6 can be proved As is wellknown it is sufficient to complete the proof ofTheorem 1 oncewe obtain Lemma 6

Lemma 6 Assume that for some 1198790gt 0 and 119861

120588(120585

0) subsub Ω one

has

(1) |1199061205850120588| + |(119883119906)

1205850120588| le 119879

0

(2) 120588 le 1205880(119879

0)

(3) Φ(1205850 120588) le Φ

0(119879

0)

Then the small conditions (49)ndash(51) are satisfied on the balls119861

120579119895120588(120585

0) for 119895 isin 119873 cup 0 Moreover the limit Λ

1205850=

lim119895rarrinfin

(119883119906)

1205850120579119895120588exists and the inequality

∮

119861120588(1205850)

1003816

1003816

1003816

1003816

1003816

119883119906 minus Λ

1205850

1003816

1003816

1003816

1003816

1003816

2

119889120585 le 119862

6((

119903

120588

)

2120591

Φ(120585

0 120588) + 119867 (119903

2

))

(69)

is valid for 0 lt 119903 le 120588 with a constant 1198626= 119862

6(119876 119873 120582 119871

120591 119886119899119889 119879

0)

Proof The proof is very similar to the proof of Lemma 51 in[12] We omit it here

Acknowledgments

The project was supported by the National Natural ScienceFoundation of China (no 11201081 and no 11126294) andby the Science and Technology Planning Project of JiangxiProvince China no GJJ13657

References

[1] J Wang and P Niu ldquoOptimal partial regularity for weaksolutions of nonlinear sub-elliptic systems in Carnot groupsrdquoNonlinear Analysis Theory Methods amp Applications vol 72 no11 pp 4162ndash4187 2010

[2] J Wang and D Liao ldquoOptimal partial regularity for sub-elliptic systems with sub-quadratic growth in Carnot groupsrdquoNonlinear Analysis Theory Methods amp Applications vol 75 no4 pp 2499ndash2519 2012

[3] E de Giorgi ldquoUn esempio di estremali discontinue per unproblema variazionale di tipo ellitticordquo Bollettino della UnioneMatematica Italiana vol 4 pp 135ndash137 1968

[4] M Giaquinta Multiple Integrals in the Calculus of Variationsand Nonlinear Elliptic Systems Princeton University PressPrinceton NJ USA 1983

[5] Y Chen and L Wu Second Order Elliptic Equations and EllipticSystems Science Press Beijing China 2003

[6] F Duzaar and K Steffen ldquoOptimal interior and boundary reg-ularity for almost minimizers to elliptic variational integralsrdquoJournal fur die Reine und Angewandte Mathematik vol 546 pp73ndash138 2002

[7] F Duzaar and J F Grotowski ldquoOptimal interior partial regu-larity for nonlinear elliptic systems the method of A-harmonicapproximationrdquo Manuscripta Mathematica vol 103 no 3 pp267ndash298 2000

[8] F Duzaar J F Grotowski and M Kronz ldquoRegularity ofalmost minimizers of quasi-convex variational integrals withsubquadratic growthrdquo Annali di Matematica Pura ed ApplicataIV vol 184 no 4 pp 421ndash448 2005

[9] F Duzaar and G Mingione ldquoThe p-harmonic approximationand the regularity of p-harmonic mapsrdquo Calculus of Variationsand Partial Differential Equations vol 20 no 3 pp 235ndash2562004

[10] F Duzaar and G Mingione ldquoRegularity for degenerate ellipticproblems via p-harmonic approximationrdquo Annales de lrsquoInstitutHenri Poincare Analyse Non Lineaire vol 21 no 5 pp 735ndash7662004

[11] S Chen andZ Tan ldquoThemethod ofA-harmonic approximationand optimal interior partial regularity for nonlinear ellipticsystems under the controllable growth conditionrdquo Journal ofMathematical Analysis and Applications vol 335 no 1 pp 20ndash42 2007

[12] F Duzaar and A Gastel ldquoNonlinear elliptic systems with Dinicontinuous coefficientsrdquo Archiv der Mathematik vol 78 no 1pp 58ndash73 2002

[13] F Duzaar A Gastel and G Mingione ldquoElliptic systemssingular sets and Dini continuityrdquo Communications in PartialDifferential Equations vol 29 no 7-8 pp 1215ndash1240 2004

[14] Y Qiu and Z Tan ldquoOptimal interior partial regularity fornonlinear elliptic systems with Dini continuous coefficientsrdquoActa Mathematica Scientia B vol 30 no 5 pp 1541ndash1554 2010

[15] Y Qiu ldquoOptimal partial regularity of second order nonlin-ear elliptic systems with Dini continuous coefficients for thesuperquadratic caserdquo Nonlinear Analysis Theory Methods ampApplications vol 75 no 8 pp 3574ndash3590 2012

[16] L Capogna ldquoRegularity of quasi-linear equations in theHeisen-berg grouprdquoCommunications on Pure andAppliedMathematicsvol 50 no 9 pp 867ndash889 1997

[17] L Capogna ldquoRegularity for quasilinear equations and 1-quasiconformal maps in Carnot groupsrdquo MathematischeAnnalen vol 313 no 2 pp 263ndash295 1999

[18] S Marchi ldquo1198621120572 local regularity for the solutions of the p-Laplacian on the Heisenberg group for 2 lt 119901 lt 1 +

radic

5rdquoZeitschrift fur Analysis und ihre Anwendungen vol 20 no 3 pp617ndash636 2001

12 Abstract and Applied Analysis

[19] S Marchi ldquo1198621120572 local regularity for the solutions of the p-Laplacian on the Heisenberg groupThe case 1+1radic5 lt 119901 le 2rdquoCommentationes Mathematicae Universitatis Carolinae vol 44no 1 pp 33ndash56 2003

[20] S Marchi ldquoL119901 regularity of the derivative in the secondcommutator direction for nonlinear elliptic equations on theHeisenberg grouprdquo Accademia Nazionale delle Scienze detta deiXL Rendiconti Serie V Memorie di Matematica e ApplicazioniParte I vol 26 pp 1ndash15 2002

[21] A Domokos ldquoDifferentiability of solutions for the non-degenerate p-Laplacian in the Heisenberg grouprdquo Journal ofDifferential Equations vol 204 no 2 pp 439ndash470 2004

[22] A Domokos On the regularity of p-harmonic functions inthe Heisenberg group [PhD thesis] University of PittsburghPittsburgh Pa USA 2004

[23] J J Manfredi and G Mingione ldquoRegularity results for quasilin-ear elliptic equations in the Heisenberg grouprdquo MathematischeAnnalen vol 339 no 3 pp 485ndash544 2007

[24] G Mingione A Zatorska-Goldstein and X Zhong ldquoGradientregularity for elliptic equations in the Heisenberg grouprdquoAdvances in Mathematics vol 222 no 1 pp 62ndash129 2009

[25] L Capogna and N Garofalo ldquoRegularity of minimizers of thecalculus of variations inCarnot groups via hypoellipticity of sys-tems ofHormander typerdquo Journal of the EuropeanMathematicalSociety vol 5 no 1 pp 1ndash40 2003

[26] E Shores ldquoHypoellipticity forlinear degenerate elliptic systemsin Carnot groups and applicationsrdquo httparxivorgabsmath0502569

[27] A Foglein ldquoPartial regularity results for subelliptic systemsin the Heisenberg grouprdquo Calculus of Variations and PartialDifferential Equations vol 32 no 1 pp 25ndash51 2008

[28] G Lu ldquoThe sharp Poincare inequality for free vector fields anendpoint resultrdquo Revista Matematica Iberoamericana vol 10no 2 pp 453ndash466 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 9: Research Article Partial Regularity for Nonlinear Subelliptic Systems …downloads.hindawi.com/journals/aaa/2013/950134.pdf · 2019-07-31 · Partial Regularity for Nonlinear Subelliptic

Abstract and Applied Analysis 9

Then we have 119883119908 = 120590

minus1

(119883119906 minus 119901

0) Now we consider

119861

120588(120585

0) subsub Ω such that 120588 le 120588

1(|119906

0| |119901

0|) Applying Lemma 4

on 119861120588(120585

0) to 119906 we have for any 120593 isin 119862infin

0(119861

120588(120585

0)R119873)

∮

119861120588(1205850)

|119883119908|

2

119889120585 = 120590

minus2

Φ(120585

0 120588 119901

0) le

1

119862

2

1

le 1 (54)

∮

119861120588(1205850)

119860

120572

119894119901119895

120573

(120585

0 119906

0 119901

0)119883119908119883120593119889120585

le [Φ

12

(120585

0 120588 119901

0)

+120596 (

1003816

1003816

1003816

1003816

119906

0

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

Φ (120585

0 120588 119901

0)) +

120575

2

] sup119861120588(1205850)

1003816

1003816

1003816

1003816

119883120593

1003816

1003816

1003816

1003816

(55)

In consideration of the small condition (49) we see that(54) and (55) imply conditions (26) in Lemma 2 Also notethat (H1) and (H3) imply condition (25) So there exists an119860

120572

119894119901119895

120573

(120585

0 119906

0 119901

0)-harmonic function ℎ isin 11986711988212(119861

120588(120585

0)R119873)

such that

∮

119861120588(1205850)

|119883ℎ|

2

119889120585 le 1 120588

minus2

∮

119861120588(1205850)

|119908 minus ℎ|

2

119889120585 le 120576 (56)

Taking 1199060= 119906

12058502120579120588 120579 isin (0 14] and replacing 119901

0by 1199010+

120590(119883ℎ)

1205850 2120579120588 we use Lemma 3 to obtain

int

119861120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119883119906 minus 119901

0minus 120590(119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

2

119889120585

le 119862

119888[

1

(120579120588)

2int

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus 119906

12058502120579120588

minus (119901

0+ 120590(119883ℎ)

12058502120579120588)

times (120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

2

119889120585 + 119865]

(57)

where

119865 = 120596

119899(2120579120588)

119876

119870

1(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119901

0+ 120590(119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 ((2120579120588)

2

)

+ [int

1198612120579120588(1205850)

(1 + 119906

119903

+ |119883119906|

2

) 119889120585]

2(1minus1119903)

(58)

Using the fact that 119906 minus (1199010+ 120590(119883ℎ)

12058502120579120588)(120585

1

minus 120585

1

0) has mean

value 11990612058502120579120588

on the ball 1198612120579120588(120585

0) the definition of119908 and (19)

we have

1

(120579120588)

2∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus 119906

1205850 2120579120588minus (119901

0+ 120590(119883ℎ)

12058502120579120588) (120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

2

119889120585

le

4120590

2

(2120579120588)

2∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119908 minus ℎ

1205850 2120579120588

minus(119883ℎ)

12058502120579120588(120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

2

119889120585

le

4120590

2

(2120579120588)

2[∮

1198612120579120588(1205850)

|119908 minus ℎ|

2

119889120585

+ ∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

ℎ minus ℎ

1205850 2120579120588

minus(119883ℎ)

12058502120579120588(120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

2

119889120585]

le 4120590

2

[(2120579)

minus119876minus2

120576 + 119862

119901∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119883ℎ minus (119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

2

119889120585]

le 4120590

2

[(2120579)

minus119876minus2

120576 + 119862

2

119901(2120579120588)

2

∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119883

2

ℎ

1003816

1003816

1003816

1003816

1003816

2

119889120585]

le 4120590

2

[(2120579)

minus119876minus2

120576 + 119862

2

119901(2120579)

2

119862

0]

le 119862

4(120579

minus119876minus2

120576 + 120579

2

) [Φ (120585

0 120588 119901

0)

+ 4120575

minus2

119870

2

1(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

) 120578 (120588

2

)]

(59)

where 1198624= 119862

4(119876 120582 119871) ge 1 Note that in the second last

inequality we have used the fact that

∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119883

2

ℎ

1003816

1003816

1003816

1003816

1003816

119889120585 le sup119861120588(1205850)

1003816

1003816

1003816

1003816

1003816

119883

2

ℎ

1003816

1003816

1003816

1003816

1003816

le 119862

0120588

minus2

∮

119861120588(1205850)

|119883ℎ|

2

119889120585 le 119862

0120588

minus2

(60)

In consideration of the fact that 119903 = 2119876(119876 minus 2) gt 2 119876 ge 4

and the assumptions 120579 isin (0 14] and Φ le 1 it follows that

[∮

1198612120579120588(1205850)

(1 + 119906

119903

+ |119883119906|

2

) 119889120585]

2(1minus1119903)

le 119862[∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

119883119906 minus 119901

0

1003816

1003816

1003816

1003816

2

119889120585]

2(1minus1119903)

+ 119862(∮

1198612120579120588(1205850)

|119883119906|

2

119889120585)

119903minus1

+ (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

4(1minus1119903)

)

10 Abstract and Applied Analysis

le 119862 [(2120579)

minus2119876(1minus1119903)

Φ(120585

0 120588 119901

0)

2(1minus1119903)

+(2120579)

minus119876(119903minus1)

Φ(120585

0 120588 119901

0)

119903minus1

]

+ (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

2(1minus1119903)

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

4(1minus1119903)

)

le 119862(2120579)

minus119876(119903minus1)

Φ(120585

0 120588 119901

0)

2(1minus1119903)

+ (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

2(1minus1119903)

)

2

(61)

Let 119875 = 1199010+120590(119883ℎ)

12058502120579120588with 119901

0= (119883119906)

12058502120579120588 Combining

these estimates (57)ndash(61) and considering the small condition(51) (it implies 120588 le 120588

1(|119906

12058502120579120588| |119875|) see (64) and (65)) we

deduce that

Φ(120585

0 120579120588) le

1003816

1003816

1003816

1003816

1003816

119861

120579120588(120585

0)

1003816

1003816

1003816

1003816

1003816

minus1

H119899int

1198612120579120588(1205850)

|119883119906 minus 119875|

2

119889120585

le 119862

119888

2

119876

(120579120588)

2∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus 119906

12058502120579120588

minus (119901

0+ 120590(119883ℎ)

12058502120579120588)

times (120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

2

119889120585

+ 2

119876

119862

119888119870

1(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119901

0+ 120590(119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

)

+ 119862

119888

(2120579120588)

2119876(1minus1119903)

(120579120588)

119876

times[∮

1198612120579120588(1205850)

(1 + 119906

119903

+ |119883119906|

2

) 119889120585]

2(1minus1119903)

le 2

119876

119862

4119862

119888(120579

minus119876minus2

120576 + 120579

2

)

times [Φ (120585

0 120588) + 4120575

minus2

119870

2

1

times (

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

) ]

+ 2

119876

119862

119888119870

1(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

+120590(119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

)

+ [2

119876

119862

119888(2120579)

2minus119876(119903minus1)

Φ(120585

0 120588)

2(1minus1119903)

+(1 +

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

2(1minus1119903)

)

2

] 120588

2

(62)

Wenow specify 120576 = 120579119876+4 120579 isin (0 14] such that 2119876+11198624119862

119888120579

2

le

120579

2120591 Note that the small condition (50) implies 12059021198625le 1 with

119862

5= max119862

0 119862

1198882

119876

(2120579)

minus(1198762+4)(119876minus2)

and it yields

2

119876

119862

119888(2120579)

2minus119876(119903minus1)

Φ(120585

0 120588)

2(1minus1119903)

le 1

(63)1003816

1003816

1003816

1003816

1003816

120590(119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

le 120590 sup1198612120579120588(1205850)

|119883ℎ|

le 120590radic119862

0(∮

119861120588(1205850)

|119883ℎ|

2

119889120585) le 120590radic119862

0le 1

(64)

where we have used the a priori estimate (28) for the A-harmonic function ℎ Furthermore using (19) and recallingthe definition of 120590 and 119862

1 we have

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

le

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588minus 119906

1205850120588

1003816

1003816

1003816

1003816

1003816

le

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+ (2120579)

minus1198762

times (∮

119861120588(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus (119883119906)

1205850120588(120585

1

minus 120585

1

0) minus 119906

1205850120588

1003816

1003816

1003816

1003816

1003816

2

119889120585)

12

le

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+ (2120579)

minus1198762

120588radic119862

119901Φ

12

(120585

0 120588)

le

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+

120590

radic

119862

119901

119862

1(2120579)

1198762

le

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+ 120590radic119862

5le

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+ 1

(65)

Combining these estimates with (62) we have

Φ(120585

0 120579120588) le 120579

2120591

Φ(120585

0 120588)

+ [4120575

minus2

119870

2

1(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

)

+ 2

119876

119862

119888119870

1(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

+120590(119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

)] 120578 (120588

2

)

+ [1 + (1 +

1003816

1003816

1003816

1003816

1003816

(119883119906)

1205850 2120579120588

1003816

1003816

1003816

1003816

1003816

2(1minus1119903)

)

2

] 120578 (120588

2

)

le 120579

2120591

Φ(120585

0 120588) + 119870 (

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

)

+ (2 +

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

)

2(119903minus1)

120578 (120588

2

)

le 120579

2120591

Φ(120585

0 120588) + 119870

lowast

(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

)

(66)

Then the proof of Lemma 5 is complete

Abstract and Applied Analysis 11

For 119879 gt 0 we find Φ0(119879) gt 0 (depending on 119876119873 120582 119871

120591 and 120596) such that

120596

2

(2119879 2Φ

0(119879)) + 2Φ

0(119879) le

1

2

120575

2

119862

1Φ

0(119879) le 120579

119876

(1 minus 120579

120591

)

2

119879

2

(67)

With Φ0(119879) from (67) we choose 120588

0(119879) isin (0 1] (depending

on 119876119873 120582 119871 120591 120596 120578 and 120581) such that

120588

0(119879) le 120588

1(1 + 2119879 1 + 2119879)

119862

3119870

2

1(2119879 2119879) 120578 (120588

0(119879)

2

) le 120575

2

119870

0(119879) 120578 (120588

0(119879)

2

) le (120579

2120574

minus 120579

2120591

)Φ

0(119879)

2 (1 + 119862

119901)119870

0(119879)119867 (120588

0(119879)

2

) le 120579

119876

(1 minus 120579

120574

)

2

(120579

2120574

minus 120579

2120591

) 119879

2

(68)

where1198700(119879) = 119870

lowast

(2119879 2119879)By the proof method of of Lemma 51 in [12] and

conditions (67)-(68) Lemma 6 can be proved As is wellknown it is sufficient to complete the proof ofTheorem 1 oncewe obtain Lemma 6

Lemma 6 Assume that for some 1198790gt 0 and 119861

120588(120585

0) subsub Ω one

has

(1) |1199061205850120588| + |(119883119906)

1205850120588| le 119879

0

(2) 120588 le 1205880(119879

0)

(3) Φ(1205850 120588) le Φ

0(119879

0)

Then the small conditions (49)ndash(51) are satisfied on the balls119861

120579119895120588(120585

0) for 119895 isin 119873 cup 0 Moreover the limit Λ

1205850=

lim119895rarrinfin

(119883119906)

1205850120579119895120588exists and the inequality

∮

119861120588(1205850)

1003816

1003816

1003816

1003816

1003816

119883119906 minus Λ

1205850

1003816

1003816

1003816

1003816

1003816

2

119889120585 le 119862

6((

119903

120588

)

2120591

Φ(120585

0 120588) + 119867 (119903

2

))

(69)

is valid for 0 lt 119903 le 120588 with a constant 1198626= 119862

6(119876 119873 120582 119871

120591 119886119899119889 119879

0)

Proof The proof is very similar to the proof of Lemma 51 in[12] We omit it here

Acknowledgments

The project was supported by the National Natural ScienceFoundation of China (no 11201081 and no 11126294) andby the Science and Technology Planning Project of JiangxiProvince China no GJJ13657

References

[1] J Wang and P Niu ldquoOptimal partial regularity for weaksolutions of nonlinear sub-elliptic systems in Carnot groupsrdquoNonlinear Analysis Theory Methods amp Applications vol 72 no11 pp 4162ndash4187 2010

[2] J Wang and D Liao ldquoOptimal partial regularity for sub-elliptic systems with sub-quadratic growth in Carnot groupsrdquoNonlinear Analysis Theory Methods amp Applications vol 75 no4 pp 2499ndash2519 2012

[3] E de Giorgi ldquoUn esempio di estremali discontinue per unproblema variazionale di tipo ellitticordquo Bollettino della UnioneMatematica Italiana vol 4 pp 135ndash137 1968

[4] M Giaquinta Multiple Integrals in the Calculus of Variationsand Nonlinear Elliptic Systems Princeton University PressPrinceton NJ USA 1983

[5] Y Chen and L Wu Second Order Elliptic Equations and EllipticSystems Science Press Beijing China 2003

[6] F Duzaar and K Steffen ldquoOptimal interior and boundary reg-ularity for almost minimizers to elliptic variational integralsrdquoJournal fur die Reine und Angewandte Mathematik vol 546 pp73ndash138 2002

[7] F Duzaar and J F Grotowski ldquoOptimal interior partial regu-larity for nonlinear elliptic systems the method of A-harmonicapproximationrdquo Manuscripta Mathematica vol 103 no 3 pp267ndash298 2000

[8] F Duzaar J F Grotowski and M Kronz ldquoRegularity ofalmost minimizers of quasi-convex variational integrals withsubquadratic growthrdquo Annali di Matematica Pura ed ApplicataIV vol 184 no 4 pp 421ndash448 2005

[9] F Duzaar and G Mingione ldquoThe p-harmonic approximationand the regularity of p-harmonic mapsrdquo Calculus of Variationsand Partial Differential Equations vol 20 no 3 pp 235ndash2562004

[10] F Duzaar and G Mingione ldquoRegularity for degenerate ellipticproblems via p-harmonic approximationrdquo Annales de lrsquoInstitutHenri Poincare Analyse Non Lineaire vol 21 no 5 pp 735ndash7662004

[11] S Chen andZ Tan ldquoThemethod ofA-harmonic approximationand optimal interior partial regularity for nonlinear ellipticsystems under the controllable growth conditionrdquo Journal ofMathematical Analysis and Applications vol 335 no 1 pp 20ndash42 2007

[12] F Duzaar and A Gastel ldquoNonlinear elliptic systems with Dinicontinuous coefficientsrdquo Archiv der Mathematik vol 78 no 1pp 58ndash73 2002

[13] F Duzaar A Gastel and G Mingione ldquoElliptic systemssingular sets and Dini continuityrdquo Communications in PartialDifferential Equations vol 29 no 7-8 pp 1215ndash1240 2004

[14] Y Qiu and Z Tan ldquoOptimal interior partial regularity fornonlinear elliptic systems with Dini continuous coefficientsrdquoActa Mathematica Scientia B vol 30 no 5 pp 1541ndash1554 2010

[15] Y Qiu ldquoOptimal partial regularity of second order nonlin-ear elliptic systems with Dini continuous coefficients for thesuperquadratic caserdquo Nonlinear Analysis Theory Methods ampApplications vol 75 no 8 pp 3574ndash3590 2012

[16] L Capogna ldquoRegularity of quasi-linear equations in theHeisen-berg grouprdquoCommunications on Pure andAppliedMathematicsvol 50 no 9 pp 867ndash889 1997

[17] L Capogna ldquoRegularity for quasilinear equations and 1-quasiconformal maps in Carnot groupsrdquo MathematischeAnnalen vol 313 no 2 pp 263ndash295 1999

[18] S Marchi ldquo1198621120572 local regularity for the solutions of the p-Laplacian on the Heisenberg group for 2 lt 119901 lt 1 +

radic

5rdquoZeitschrift fur Analysis und ihre Anwendungen vol 20 no 3 pp617ndash636 2001

12 Abstract and Applied Analysis

[19] S Marchi ldquo1198621120572 local regularity for the solutions of the p-Laplacian on the Heisenberg groupThe case 1+1radic5 lt 119901 le 2rdquoCommentationes Mathematicae Universitatis Carolinae vol 44no 1 pp 33ndash56 2003

[20] S Marchi ldquoL119901 regularity of the derivative in the secondcommutator direction for nonlinear elliptic equations on theHeisenberg grouprdquo Accademia Nazionale delle Scienze detta deiXL Rendiconti Serie V Memorie di Matematica e ApplicazioniParte I vol 26 pp 1ndash15 2002

[21] A Domokos ldquoDifferentiability of solutions for the non-degenerate p-Laplacian in the Heisenberg grouprdquo Journal ofDifferential Equations vol 204 no 2 pp 439ndash470 2004

[22] A Domokos On the regularity of p-harmonic functions inthe Heisenberg group [PhD thesis] University of PittsburghPittsburgh Pa USA 2004

[23] J J Manfredi and G Mingione ldquoRegularity results for quasilin-ear elliptic equations in the Heisenberg grouprdquo MathematischeAnnalen vol 339 no 3 pp 485ndash544 2007

[24] G Mingione A Zatorska-Goldstein and X Zhong ldquoGradientregularity for elliptic equations in the Heisenberg grouprdquoAdvances in Mathematics vol 222 no 1 pp 62ndash129 2009

[25] L Capogna and N Garofalo ldquoRegularity of minimizers of thecalculus of variations inCarnot groups via hypoellipticity of sys-tems ofHormander typerdquo Journal of the EuropeanMathematicalSociety vol 5 no 1 pp 1ndash40 2003

[26] E Shores ldquoHypoellipticity forlinear degenerate elliptic systemsin Carnot groups and applicationsrdquo httparxivorgabsmath0502569

[27] A Foglein ldquoPartial regularity results for subelliptic systemsin the Heisenberg grouprdquo Calculus of Variations and PartialDifferential Equations vol 32 no 1 pp 25ndash51 2008

[28] G Lu ldquoThe sharp Poincare inequality for free vector fields anendpoint resultrdquo Revista Matematica Iberoamericana vol 10no 2 pp 453ndash466 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 10: Research Article Partial Regularity for Nonlinear Subelliptic Systems …downloads.hindawi.com/journals/aaa/2013/950134.pdf · 2019-07-31 · Partial Regularity for Nonlinear Subelliptic

10 Abstract and Applied Analysis

le 119862 [(2120579)

minus2119876(1minus1119903)

Φ(120585

0 120588 119901

0)

2(1minus1119903)

+(2120579)

minus119876(119903minus1)

Φ(120585

0 120588 119901

0)

119903minus1

]

+ (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

2(1minus1119903)

+

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

4(1minus1119903)

)

le 119862(2120579)

minus119876(119903minus1)

Φ(120585

0 120588 119901

0)

2(1minus1119903)

+ (1 +

1003816

1003816

1003816

1003816

119901

0

1003816

1003816

1003816

1003816

2(1minus1119903)

)

2

(61)

Let 119875 = 1199010+120590(119883ℎ)

12058502120579120588with 119901

0= (119883119906)

12058502120579120588 Combining

these estimates (57)ndash(61) and considering the small condition(51) (it implies 120588 le 120588

1(|119906

12058502120579120588| |119875|) see (64) and (65)) we

deduce that

Φ(120585

0 120579120588) le

1003816

1003816

1003816

1003816

1003816

119861

120579120588(120585

0)

1003816

1003816

1003816

1003816

1003816

minus1

H119899int

1198612120579120588(1205850)

|119883119906 minus 119875|

2

119889120585

le 119862

119888

2

119876

(120579120588)

2∮

1198612120579120588(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus 119906

12058502120579120588

minus (119901

0+ 120590(119883ℎ)

12058502120579120588)

times (120585

1

minus 120585

1

0)

1003816

1003816

1003816

1003816

1003816

2

119889120585

+ 2

119876

119862

119888119870

1(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119901

0+ 120590(119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

)

+ 119862

119888

(2120579120588)

2119876(1minus1119903)

(120579120588)

119876

times[∮

1198612120579120588(1205850)

(1 + 119906

119903

+ |119883119906|

2

) 119889120585]

2(1minus1119903)

le 2

119876

119862

4119862

119888(120579

minus119876minus2

120576 + 120579

2

)

times [Φ (120585

0 120588) + 4120575

minus2

119870

2

1

times (

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

) ]

+ 2

119876

119862

119888119870

1(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

+120590(119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

)

+ [2

119876

119862

119888(2120579)

2minus119876(119903minus1)

Φ(120585

0 120588)

2(1minus1119903)

+(1 +

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

2(1minus1119903)

)

2

] 120588

2

(62)

Wenow specify 120576 = 120579119876+4 120579 isin (0 14] such that 2119876+11198624119862

119888120579

2

le

120579

2120591 Note that the small condition (50) implies 12059021198625le 1 with

119862

5= max119862

0 119862

1198882

119876

(2120579)

minus(1198762+4)(119876minus2)

and it yields

2

119876

119862

119888(2120579)

2minus119876(119903minus1)

Φ(120585

0 120588)

2(1minus1119903)

le 1

(63)1003816

1003816

1003816

1003816

1003816

120590(119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

le 120590 sup1198612120579120588(1205850)

|119883ℎ|

le 120590radic119862

0(∮

119861120588(1205850)

|119883ℎ|

2

119889120585) le 120590radic119862

0le 1

(64)

where we have used the a priori estimate (28) for the A-harmonic function ℎ Furthermore using (19) and recallingthe definition of 120590 and 119862

1 we have

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

le

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588minus 119906

1205850120588

1003816

1003816

1003816

1003816

1003816

le

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+ (2120579)

minus1198762

times (∮

119861120588(1205850)

1003816

1003816

1003816

1003816

1003816

119906 minus (119883119906)

1205850120588(120585

1

minus 120585

1

0) minus 119906

1205850120588

1003816

1003816

1003816

1003816

1003816

2

119889120585)

12

le

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+ (2120579)

minus1198762

120588radic119862

119901Φ

12

(120585

0 120588)

le

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+

120590

radic

119862

119901

119862

1(2120579)

1198762

le

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+ 120590radic119862

5le

1003816

1003816

1003816

1003816

1003816

119906

1205850120588

1003816

1003816

1003816

1003816

1003816

+ 1

(65)

Combining these estimates with (62) we have

Φ(120585

0 120579120588) le 120579

2120591

Φ(120585

0 120588)

+ [4120575

minus2

119870

2

1(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

)

+ 2

119876

119862

119888119870

1(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

+120590(119883ℎ)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

)] 120578 (120588

2

)

+ [1 + (1 +

1003816

1003816

1003816

1003816

1003816

(119883119906)

1205850 2120579120588

1003816

1003816

1003816

1003816

1003816

2(1minus1119903)

)

2

] 120578 (120588

2

)

le 120579

2120591

Φ(120585

0 120588) + 119870 (

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

)

+ (2 +

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

)

2(119903minus1)

120578 (120588

2

)

le 120579

2120591

Φ(120585

0 120588) + 119870

lowast

(

1003816

1003816

1003816

1003816

1003816

119906

12058502120579120588

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(119883119906)

12058502120579120588

1003816

1003816

1003816

1003816

1003816

) 120578 (120588

2

)

(66)

Then the proof of Lemma 5 is complete

Abstract and Applied Analysis 11

For 119879 gt 0 we find Φ0(119879) gt 0 (depending on 119876119873 120582 119871

120591 and 120596) such that

120596

2

(2119879 2Φ

0(119879)) + 2Φ

0(119879) le

1

2

120575

2

119862

1Φ

0(119879) le 120579

119876

(1 minus 120579

120591

)

2

119879

2

(67)

With Φ0(119879) from (67) we choose 120588

0(119879) isin (0 1] (depending

on 119876119873 120582 119871 120591 120596 120578 and 120581) such that

120588

0(119879) le 120588

1(1 + 2119879 1 + 2119879)

119862

3119870

2

1(2119879 2119879) 120578 (120588

0(119879)

2

) le 120575

2

119870

0(119879) 120578 (120588

0(119879)

2

) le (120579

2120574

minus 120579

2120591

)Φ

0(119879)

2 (1 + 119862

119901)119870

0(119879)119867 (120588

0(119879)

2

) le 120579

119876

(1 minus 120579

120574

)

2

(120579

2120574

minus 120579

2120591

) 119879

2

(68)

where1198700(119879) = 119870

lowast

(2119879 2119879)By the proof method of of Lemma 51 in [12] and

conditions (67)-(68) Lemma 6 can be proved As is wellknown it is sufficient to complete the proof ofTheorem 1 oncewe obtain Lemma 6

Lemma 6 Assume that for some 1198790gt 0 and 119861

120588(120585

0) subsub Ω one

has

(1) |1199061205850120588| + |(119883119906)

1205850120588| le 119879

0

(2) 120588 le 1205880(119879

0)

(3) Φ(1205850 120588) le Φ

0(119879

0)

Then the small conditions (49)ndash(51) are satisfied on the balls119861

120579119895120588(120585

0) for 119895 isin 119873 cup 0 Moreover the limit Λ

1205850=

lim119895rarrinfin

(119883119906)

1205850120579119895120588exists and the inequality

∮

119861120588(1205850)

1003816

1003816

1003816

1003816

1003816

119883119906 minus Λ

1205850

1003816

1003816

1003816

1003816

1003816

2

119889120585 le 119862

6((

119903

120588

)

2120591

Φ(120585

0 120588) + 119867 (119903

2

))

(69)

is valid for 0 lt 119903 le 120588 with a constant 1198626= 119862

6(119876 119873 120582 119871

120591 119886119899119889 119879

0)

Proof The proof is very similar to the proof of Lemma 51 in[12] We omit it here

Acknowledgments

The project was supported by the National Natural ScienceFoundation of China (no 11201081 and no 11126294) andby the Science and Technology Planning Project of JiangxiProvince China no GJJ13657

References

[1] J Wang and P Niu ldquoOptimal partial regularity for weaksolutions of nonlinear sub-elliptic systems in Carnot groupsrdquoNonlinear Analysis Theory Methods amp Applications vol 72 no11 pp 4162ndash4187 2010

[2] J Wang and D Liao ldquoOptimal partial regularity for sub-elliptic systems with sub-quadratic growth in Carnot groupsrdquoNonlinear Analysis Theory Methods amp Applications vol 75 no4 pp 2499ndash2519 2012

[3] E de Giorgi ldquoUn esempio di estremali discontinue per unproblema variazionale di tipo ellitticordquo Bollettino della UnioneMatematica Italiana vol 4 pp 135ndash137 1968

[4] M Giaquinta Multiple Integrals in the Calculus of Variationsand Nonlinear Elliptic Systems Princeton University PressPrinceton NJ USA 1983

[5] Y Chen and L Wu Second Order Elliptic Equations and EllipticSystems Science Press Beijing China 2003

[6] F Duzaar and K Steffen ldquoOptimal interior and boundary reg-ularity for almost minimizers to elliptic variational integralsrdquoJournal fur die Reine und Angewandte Mathematik vol 546 pp73ndash138 2002

[7] F Duzaar and J F Grotowski ldquoOptimal interior partial regu-larity for nonlinear elliptic systems the method of A-harmonicapproximationrdquo Manuscripta Mathematica vol 103 no 3 pp267ndash298 2000

[8] F Duzaar J F Grotowski and M Kronz ldquoRegularity ofalmost minimizers of quasi-convex variational integrals withsubquadratic growthrdquo Annali di Matematica Pura ed ApplicataIV vol 184 no 4 pp 421ndash448 2005

[9] F Duzaar and G Mingione ldquoThe p-harmonic approximationand the regularity of p-harmonic mapsrdquo Calculus of Variationsand Partial Differential Equations vol 20 no 3 pp 235ndash2562004

[10] F Duzaar and G Mingione ldquoRegularity for degenerate ellipticproblems via p-harmonic approximationrdquo Annales de lrsquoInstitutHenri Poincare Analyse Non Lineaire vol 21 no 5 pp 735ndash7662004

[11] S Chen andZ Tan ldquoThemethod ofA-harmonic approximationand optimal interior partial regularity for nonlinear ellipticsystems under the controllable growth conditionrdquo Journal ofMathematical Analysis and Applications vol 335 no 1 pp 20ndash42 2007

[12] F Duzaar and A Gastel ldquoNonlinear elliptic systems with Dinicontinuous coefficientsrdquo Archiv der Mathematik vol 78 no 1pp 58ndash73 2002

[13] F Duzaar A Gastel and G Mingione ldquoElliptic systemssingular sets and Dini continuityrdquo Communications in PartialDifferential Equations vol 29 no 7-8 pp 1215ndash1240 2004

[14] Y Qiu and Z Tan ldquoOptimal interior partial regularity fornonlinear elliptic systems with Dini continuous coefficientsrdquoActa Mathematica Scientia B vol 30 no 5 pp 1541ndash1554 2010

[15] Y Qiu ldquoOptimal partial regularity of second order nonlin-ear elliptic systems with Dini continuous coefficients for thesuperquadratic caserdquo Nonlinear Analysis Theory Methods ampApplications vol 75 no 8 pp 3574ndash3590 2012

[16] L Capogna ldquoRegularity of quasi-linear equations in theHeisen-berg grouprdquoCommunications on Pure andAppliedMathematicsvol 50 no 9 pp 867ndash889 1997

[17] L Capogna ldquoRegularity for quasilinear equations and 1-quasiconformal maps in Carnot groupsrdquo MathematischeAnnalen vol 313 no 2 pp 263ndash295 1999

[18] S Marchi ldquo1198621120572 local regularity for the solutions of the p-Laplacian on the Heisenberg group for 2 lt 119901 lt 1 +

radic

5rdquoZeitschrift fur Analysis und ihre Anwendungen vol 20 no 3 pp617ndash636 2001

12 Abstract and Applied Analysis

[19] S Marchi ldquo1198621120572 local regularity for the solutions of the p-Laplacian on the Heisenberg groupThe case 1+1radic5 lt 119901 le 2rdquoCommentationes Mathematicae Universitatis Carolinae vol 44no 1 pp 33ndash56 2003

[20] S Marchi ldquoL119901 regularity of the derivative in the secondcommutator direction for nonlinear elliptic equations on theHeisenberg grouprdquo Accademia Nazionale delle Scienze detta deiXL Rendiconti Serie V Memorie di Matematica e ApplicazioniParte I vol 26 pp 1ndash15 2002

[21] A Domokos ldquoDifferentiability of solutions for the non-degenerate p-Laplacian in the Heisenberg grouprdquo Journal ofDifferential Equations vol 204 no 2 pp 439ndash470 2004

[22] A Domokos On the regularity of p-harmonic functions inthe Heisenberg group [PhD thesis] University of PittsburghPittsburgh Pa USA 2004

[23] J J Manfredi and G Mingione ldquoRegularity results for quasilin-ear elliptic equations in the Heisenberg grouprdquo MathematischeAnnalen vol 339 no 3 pp 485ndash544 2007

[24] G Mingione A Zatorska-Goldstein and X Zhong ldquoGradientregularity for elliptic equations in the Heisenberg grouprdquoAdvances in Mathematics vol 222 no 1 pp 62ndash129 2009

[25] L Capogna and N Garofalo ldquoRegularity of minimizers of thecalculus of variations inCarnot groups via hypoellipticity of sys-tems ofHormander typerdquo Journal of the EuropeanMathematicalSociety vol 5 no 1 pp 1ndash40 2003

[26] E Shores ldquoHypoellipticity forlinear degenerate elliptic systemsin Carnot groups and applicationsrdquo httparxivorgabsmath0502569

[27] A Foglein ldquoPartial regularity results for subelliptic systemsin the Heisenberg grouprdquo Calculus of Variations and PartialDifferential Equations vol 32 no 1 pp 25ndash51 2008

[28] G Lu ldquoThe sharp Poincare inequality for free vector fields anendpoint resultrdquo Revista Matematica Iberoamericana vol 10no 2 pp 453ndash466 1994

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Differential EquationsInternational Journal of

Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 11: Research Article Partial Regularity for Nonlinear Subelliptic Systems …downloads.hindawi.com/journals/aaa/2013/950134.pdf · 2019-07-31 · Partial Regularity for Nonlinear Subelliptic

Abstract and Applied Analysis 11

For 119879 gt 0 we find Φ0(119879) gt 0 (depending on 119876119873 120582 119871

120591 and 120596) such that

120596

2

(2119879 2Φ

0(119879)) + 2Φ

0(119879) le

1

2

120575

2

119862

1Φ

0(119879) le 120579

119876

(1 minus 120579

120591

)

2

119879

2

(67)

With Φ0(119879) from (67) we choose 120588

0(119879) isin (0 1] (depending

on 119876119873 120582 119871 120591 120596 120578 and 120581) such that

120588

0(119879) le 120588

1(1 + 2119879 1 + 2119879)

119862

3119870

2

1(2119879 2119879) 120578 (120588

0(119879)

2

) le 120575

2

119870

0(119879) 120578 (120588

0(119879)

2

) le (120579

2120574

minus 120579

2120591

)Φ

0(119879)

2 (1 + 119862

119901)119870

0(119879)119867 (120588

0(119879)

2

) le 120579

119876

(1 minus 120579

120574

)

2

(120579

2120574

minus 120579

2120591

) 119879

2

(68)

where1198700(119879) = 119870

lowast

(2119879 2119879)By the proof method of of Lemma 51 in [12] and

conditions (67)-(68) Lemma 6 can be proved As is wellknown it is sufficient to complete the proof ofTheorem 1 oncewe obtain Lemma 6

Lemma 6 Assume that for some 1198790gt 0 and 119861

120588(120585

0) subsub Ω one

has

(1) |1199061205850120588| + |(119883119906)

1205850120588| le 119879

0

(2) 120588 le 1205880(119879

0)

(3) Φ(1205850 120588) le Φ

0(119879

0)

Then the small conditions (49)ndash(51) are satisfied on the balls119861

120579119895120588(120585

0) for 119895 isin 119873 cup 0 Moreover the limit Λ

1205850=

lim119895rarrinfin

(119883119906)

1205850120579119895120588exists and the inequality

∮

119861120588(1205850)

1003816

1003816

1003816

1003816

1003816

119883119906 minus Λ

1205850

1003816

1003816

1003816

1003816

1003816

2

119889120585 le 119862

6((

119903

120588

)

2120591

Φ(120585

0 120588) + 119867 (119903

2

))

(69)

is valid for 0 lt 119903 le 120588 with a constant 1198626= 119862

6(119876 119873 120582 119871

120591 119886119899119889 119879

0)

Proof The proof is very similar to the proof of Lemma 51 in[12] We omit it here

Acknowledgments

The project was supported by the National Natural ScienceFoundation of China (no 11201081 and no 11126294) andby the Science and Technology Planning Project of JiangxiProvince China no GJJ13657

References

[1] J Wang and P Niu ldquoOptimal partial regularity for weaksolutions of nonlinear sub-elliptic systems in Carnot groupsrdquoNonlinear Analysis Theory Methods amp Applications vol 72 no11 pp 4162ndash4187 2010

[2] J Wang and D Liao ldquoOptimal partial regularity for sub-elliptic systems with sub-quadratic growth in Carnot groupsrdquoNonlinear Analysis Theory Methods amp Applications vol 75 no4 pp 2499ndash2519 2012

[3] E de Giorgi ldquoUn esempio di estremali discontinue per unproblema variazionale di tipo ellitticordquo Bollettino della UnioneMatematica Italiana vol 4 pp 135ndash137 1968

[4] M Giaquinta Multiple Integrals in the Calculus of Variationsand Nonlinear Elliptic Systems Princeton University PressPrinceton NJ USA 1983

[5] Y Chen and L Wu Second Order Elliptic Equations and EllipticSystems Science Press Beijing China 2003

[6] F Duzaar and K Steffen ldquoOptimal interior and boundary reg-ularity for almost minimizers to elliptic variational integralsrdquoJournal fur die Reine und Angewandte Mathematik vol 546 pp73ndash138 2002

[7] F Duzaar and J F Grotowski ldquoOptimal interior partial regu-larity for nonlinear elliptic systems the method of A-harmonicapproximationrdquo Manuscripta Mathematica vol 103 no 3 pp267ndash298 2000

[8] F Duzaar J F Grotowski and M Kronz ldquoRegularity ofalmost minimizers of quasi-convex variational integrals withsubquadratic growthrdquo Annali di Matematica Pura ed ApplicataIV vol 184 no 4 pp 421ndash448 2005

[9] F Duzaar and G Mingione ldquoThe p-harmonic approximationand the regularity of p-harmonic mapsrdquo Calculus of Variationsand Partial Differential Equations vol 20 no 3 pp 235ndash2562004

[10] F Duzaar and G Mingione ldquoRegularity for degenerate ellipticproblems via p-harmonic approximationrdquo Annales de lrsquoInstitutHenri Poincare Analyse Non Lineaire vol 21 no 5 pp 735ndash7662004

[11] S Chen andZ Tan ldquoThemethod ofA-harmonic approximationand optimal interior partial regularity for nonlinear ellipticsystems under the controllable growth conditionrdquo Journal ofMathematical Analysis and Applications vol 335 no 1 pp 20ndash42 2007

[12] F Duzaar and A Gastel ldquoNonlinear elliptic systems with Dinicontinuous coefficientsrdquo Archiv der Mathematik vol 78 no 1pp 58ndash73 2002

[13] F Duzaar A Gastel and G Mingione ldquoElliptic systemssingular sets and Dini continuityrdquo Communications in PartialDifferential Equations vol 29 no 7-8 pp 1215ndash1240 2004

[14] Y Qiu and Z Tan ldquoOptimal interior partial regularity fornonlinear elliptic systems with Dini continuous coefficientsrdquoActa Mathematica Scientia B vol 30 no 5 pp 1541ndash1554 2010

[15] Y Qiu ldquoOptimal partial regularity of second order nonlin-ear elliptic systems with Dini continuous coefficients for thesuperquadratic caserdquo Nonlinear Analysis Theory Methods ampApplications vol 75 no 8 pp 3574ndash3590 2012

[16] L Capogna ldquoRegularity of quasi-linear equations in theHeisen-berg grouprdquoCommunications on Pure andAppliedMathematicsvol 50 no 9 pp 867ndash889 1997

[17] L Capogna ldquoRegularity for quasilinear equations and 1-quasiconformal maps in Carnot groupsrdquo MathematischeAnnalen vol 313 no 2 pp 263ndash295 1999

[18] S Marchi ldquo1198621120572 local regularity for the solutions of the p-Laplacian on the Heisenberg group for 2 lt 119901 lt 1 +

radic

5rdquoZeitschrift fur Analysis und ihre Anwendungen vol 20 no 3 pp617ndash636 2001

12 Abstract and Applied Analysis

[19] S Marchi ldquo1198621120572 local regularity for the solutions of the p-Laplacian on the Heisenberg groupThe case 1+1radic5 lt 119901 le 2rdquoCommentationes Mathematicae Universitatis Carolinae vol 44no 1 pp 33ndash56 2003

[20] S Marchi ldquoL119901 regularity of the derivative in the secondcommutator direction for nonlinear elliptic equations on theHeisenberg grouprdquo Accademia Nazionale delle Scienze detta deiXL Rendiconti Serie V Memorie di Matematica e ApplicazioniParte I vol 26 pp 1ndash15 2002

[21] A Domokos ldquoDifferentiability of solutions for the non-degenerate p-Laplacian in the Heisenberg grouprdquo Journal ofDifferential Equations vol 204 no 2 pp 439ndash470 2004

[22] A Domokos On the regularity of p-harmonic functions inthe Heisenberg group [PhD thesis] University of PittsburghPittsburgh Pa USA 2004

[23] J J Manfredi and G Mingione ldquoRegularity results for quasilin-ear elliptic equations in the Heisenberg grouprdquo MathematischeAnnalen vol 339 no 3 pp 485ndash544 2007

[24] G Mingione A Zatorska-Goldstein and X Zhong ldquoGradientregularity for elliptic equations in the Heisenberg grouprdquoAdvances in Mathematics vol 222 no 1 pp 62ndash129 2009

[25] L Capogna and N Garofalo ldquoRegularity of minimizers of thecalculus of variations inCarnot groups via hypoellipticity of sys-tems ofHormander typerdquo Journal of the EuropeanMathematicalSociety vol 5 no 1 pp 1ndash40 2003

[26] E Shores ldquoHypoellipticity forlinear degenerate elliptic systemsin Carnot groups and applicationsrdquo httparxivorgabsmath0502569

[27] A Foglein ldquoPartial regularity results for subelliptic systemsin the Heisenberg grouprdquo Calculus of Variations and PartialDifferential Equations vol 32 no 1 pp 25ndash51 2008

[28] G Lu ldquoThe sharp Poincare inequality for free vector fields anendpoint resultrdquo Revista Matematica Iberoamericana vol 10no 2 pp 453ndash466 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 12: Research Article Partial Regularity for Nonlinear Subelliptic Systems …downloads.hindawi.com/journals/aaa/2013/950134.pdf · 2019-07-31 · Partial Regularity for Nonlinear Subelliptic

12 Abstract and Applied Analysis

[19] S Marchi ldquo1198621120572 local regularity for the solutions of the p-Laplacian on the Heisenberg groupThe case 1+1radic5 lt 119901 le 2rdquoCommentationes Mathematicae Universitatis Carolinae vol 44no 1 pp 33ndash56 2003

[20] S Marchi ldquoL119901 regularity of the derivative in the secondcommutator direction for nonlinear elliptic equations on theHeisenberg grouprdquo Accademia Nazionale delle Scienze detta deiXL Rendiconti Serie V Memorie di Matematica e ApplicazioniParte I vol 26 pp 1ndash15 2002

[21] A Domokos ldquoDifferentiability of solutions for the non-degenerate p-Laplacian in the Heisenberg grouprdquo Journal ofDifferential Equations vol 204 no 2 pp 439ndash470 2004

[22] A Domokos On the regularity of p-harmonic functions inthe Heisenberg group [PhD thesis] University of PittsburghPittsburgh Pa USA 2004

[23] J J Manfredi and G Mingione ldquoRegularity results for quasilin-ear elliptic equations in the Heisenberg grouprdquo MathematischeAnnalen vol 339 no 3 pp 485ndash544 2007

[24] G Mingione A Zatorska-Goldstein and X Zhong ldquoGradientregularity for elliptic equations in the Heisenberg grouprdquoAdvances in Mathematics vol 222 no 1 pp 62ndash129 2009

[25] L Capogna and N Garofalo ldquoRegularity of minimizers of thecalculus of variations inCarnot groups via hypoellipticity of sys-tems ofHormander typerdquo Journal of the EuropeanMathematicalSociety vol 5 no 1 pp 1ndash40 2003

[26] E Shores ldquoHypoellipticity forlinear degenerate elliptic systemsin Carnot groups and applicationsrdquo httparxivorgabsmath0502569

[27] A Foglein ldquoPartial regularity results for subelliptic systemsin the Heisenberg grouprdquo Calculus of Variations and PartialDifferential Equations vol 32 no 1 pp 25ndash51 2008

[28] G Lu ldquoThe sharp Poincare inequality for free vector fields anendpoint resultrdquo Revista Matematica Iberoamericana vol 10no 2 pp 453ndash466 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 13: Research Article Partial Regularity for Nonlinear Subelliptic Systems …downloads.hindawi.com/journals/aaa/2013/950134.pdf · 2019-07-31 · Partial Regularity for Nonlinear Subelliptic

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


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