Besov regularity of solutions of the p-Laplace …Besov regularity of solutions of the p-Laplace...

Besov regularity of solutions of the p-Laplace equation

Benjamin Scharf

Technische Universitat Munchen,Department of Mathematics,Applied Numerical Analysis

[email protected]

joint work with Lars Diening (Munich),Stephan Dahlke, Christoph Hartmann, Markus Weimar (Marburg)

Jena, June 27, 2014

Overview

Introduction and results for the Laplace equation (p = 2)Introduction to the p-LaplaceApproximation in Sobolev and Besov spacesKnown results for the Laplace equation (p = 2)

Sobolev and local Holder regularity of the p-LaplaceSobolev regularity of the p-LaplaceLocal Holder regularity of the p-Laplace equation

Besov regularity of solutions of the p-Laplace equationFrom Bs

p,p(Ω) and C `,αγ,loc(Ω) to Bσ

τ,τ (Ω)Besov regularity of the p-Laplace

Benjamin Scharf Besov regularity of solutions of the p-Laplace equation 2 of 23

Introduction and known results – Introduction to the p-Laplace

The p-Laplace - Introduction

Ω ⊂ Rd Lipschitz domain, d dimension, 1 < p <∞

Inhomogeneous p-Laplace equation:

∆pu := div(|∇u|p−2∇u

)= f in Ω, u = 0 on ∂Ω.

Variational (weak) formulation:∫Ω

⟨|∇u|p−2∇u,∇v

⟩dx =

∫Ω

f v dx for all v ∈ C∞0 (Ω)

has a unique solution u ∈ W 1p (Ω) for f ∈W−1

p′ (Ω),

has model character for nonlinear problems, similar to the Laplaceequation (p = 2) for linear problems

nice and free introduction: P. Lindqvist. Notes on the p-Laplace equation, 2006.

http: // www. math. ntnu. no/ ~ lqvist/ p-laplace. pdf


http://www.math.ntnu.no/~lqvist/p-laplace.pdf

Introduction and known results – Approximation in Sobolev and Besov spaces

Sobolev and Besov spaces

W sp (Ω): Sobolev space of smoothness s and integrability p on Ω

Bsp,p(Ω): Besov space of smoothness s and integrability p on Ω

Wavelet representation: ηI ,p = |I |1/2−1/p ηI p-normalized wavelets

g ∈ Bsp,p(Rd )⇔ g = P0(g) +

∑I

∑η∈Ψ

⟨g , ηI ,p′

⟩ηI ,p

and∥∥∥P0(g) Lp(Rd )

∥∥∥+∥∥∥⟨g , ηI ,p′

⟩bs

p,p(Rd )∥∥∥ <∞

Here ∥∥∥⟨g , ηI ,p′⟩

bsp,p(Rd )

∥∥∥p=∑

I

∑η∈Ψ

|I |−sp/d∣∣⟨g , ηI ,p′

⟩∣∣pmore smoothness ⇔ more decay of the wavelet coefficients

Trivial embedding: Bs+εp,p (Ω) →W s

p (Ω) → Bsp,p(Ω)



Sobolev and Besov spaces

W sp (Ω): Sobolev space of smoothness s and integrability p on Ω

Bsp,p(Ω): Besov space of smoothness s and integrability p on Ω

Wavelet representation: ηI ,p = |I |1/2−1/p ηI p-normalized wavelets

g ∈ Bsp,p(Rd )⇔ g = P0(g) +

∑I

∑η∈Ψ

⟨g , ηI ,p′

⟩ηI ,p

and∥∥∥P0(g) Lp(Rd )

∥∥∥+∥∥∥⟨g , ηI ,p′

⟩bs

p,p(Rd )∥∥∥ <∞

Here ∥∥∥⟨g , ηI ,p′⟩

bsp,p(Rd )

∥∥∥p=∑

I

∑η∈Ψ

|I |−sp/d∣∣⟨g , ηI ,p′

⟩∣∣pmore smoothness ⇔ more decay of the wavelet coefficients

Trivial embedding: Bs+εp,p (Ω) →W s

p (Ω) → Bsp,p(Ω)



Linear and Adaptive approximation by wavelets (i)

How to approximate f ∈ Bsp,p(Ω), Ω bounded, by wavelet basis?

Linear approximation fk of f (order k : ∼ 2kd terms):

fk = P0(g) +∑|I |≥2−k

∑η∈Ψ

⟨g , ηI ,p′

⟩ηI ,p

It holds

f ∈ Bsp,p(Ω) (or W s

p (Ω))⇒ ‖f − fk Lp(Ω)‖ . 2−ks .



Linear and Adaptive approximation by wavelets (ii)

Adaptive approximation fk of f (order k : ∼ 2kd terms):

f Dk = P0(g) +

∑(I ,η)∈D

⟨g , ηI ,p′

⟩ηI ,p with |D| = 2kd

best m-term approximation: choose D to minimize∥∥∥f − f Dk Lp(Ω)

∥∥∥ : take 2kd largest wavelet coefficients!

Let 1τ = σ

d + 1p , in particular τ < 1 possible. It holds

f ∈ Bστ,τ (Ω)⇒ ‖f − fk Lp(Ω)‖ ∼ 2−kσ

Besov regularity is the maximal possible convergence rate of anadaptive algorithm ⇒ how much higher than Sobolev regularity?



Linear and Adaptive approximation by wavelets (iii)

The main reason is the following computation:

TheoremLet 1

τ = σd + 1

p , x ∈ `τ and x∗ its non-increasing rearrangement. Then

‖x∗ − x∗k‖p ≤ k−σd ‖x‖τ ,

where x∗k is the cut-off of x∗ after the k first terms.

Proof:Assume w.l.o.g. that ‖x‖τ = 1. Then

|x∗(j)|τ ≤ |x∗(k)|τ ≤ 1

k‖x∗‖ττ =

1

k· for j > k .

Therefore

‖x∗ − x∗k‖pp ≤ ‖x∗ − x∗k‖p−τ

∞ · ‖x∗ − x∗k‖ττ ≤ kτ−pτ · 1 = k−

σd

p.


Introduction and known results – Known results for the Laplace equation (p = 2)

Sobolev regularity for p = 2, the linear case

Theorem (Jerison, Kenig 1981,1995, Theorem B)

Positive: Lipschitz domain Ω ∈ Rd , f ∈ L2(Ω). Then the solution u of

∆u = f in Ω, u = 0 on ∂Ω

belongs to W3/22 (Ω).

Negative: For any s > 3/2 there exists a Lipschitz domain Ω andsmooth f s.t. u with

∆u = f in Ω, u = 0 on ∂Ω

does not belong to W s2 (Ω).

Careful! ∃ C1-domain Ω and f ∈W−1/22 (Ω) such that u /∈W

3/22 (Ω)

D. Jerison, C.E. Kenig. The Inhomogeneous Dirichlet Problem in Lipschitz Domains. J.Funct. Anal. 130, 161–219, 1995.



Besov regularity for p = 2 (i)

Theorem (Dahlke,DeVore ’97; Jerison,Kenig ’95; Hansen 2013)

Lipschitz domain Ω ∈ Rd , f ∈W γ2 (Ω) for γ ≥ max

(4−d

2d−2 , 0)

. Then

the solution u of

∆u = f in Ω, u = 0 on ∂Ω

belongs to Bστ,τ (Ω), 1

τ = σd + 1

p , for any σ < 32 ·

dd−1 .

Besov reg. always better than 3/2, the maximal Sobolev regularity proof by a general embedding:

small global Sobolev regularity + better local (weighted) Sobolevregularity (Babuska-Kondratiev) result in better Besov regularity!

S. Dahlke, R.A. DeVore. Besov regularity for elliptic boundary value problems. Comm.Partial Differential Equations, 22(1–2), 1–16, 1997.

M. Hansen, n-term approximation rates and Besov regularity for elliptic PDEs onpolyhedral domains, to appear in J. Found. Comp. Math.



Besov regularity for p = 2 (i)

Theorem (Dahlke,DeVore ’97; Jerison,Kenig ’95; Hansen 2013)

Lipschitz domain Ω ∈ Rd , f ∈W γ2 (Ω) for γ ≥ max

(4−d

2d−2 , 0)

. Then

the solution u of

∆u = f in Ω, u = 0 on ∂Ω

belongs to Bστ,τ (Ω), 1

τ = σd + 1

p , for any σ < 32 ·

dd−1 .

Besov reg. always better than 3/2, the maximal Sobolev regularity proof by a general embedding:

small global Sobolev regularity + better local (weighted) Sobolevregularity (Babuska-Kondratiev) result in better Besov regularity!

S. Dahlke, R.A. DeVore. Besov regularity for elliptic boundary value problems. Comm.Partial Differential Equations, 22(1–2), 1–16, 1997.

M. Hansen, n-term approximation rates and Besov regularity for elliptic PDEs onpolyhedral domains, to appear in J. Found. Comp. Math.



Besov regularity for p = 2 (ii)

Proof Idea:

extend u to Rn and take its wavelet decomposition – 3 parts

1. father wavelets (independent of regularity)2. interior and exterior wavelets ηI ,p with

dist(I , ∂Ω) & diam(I ) (1)

3. boundary wavelets ηI ,p; (1) doesn’t hold

handle 3 parts separately

1. no problem2. use weighted Sobolev reg.: If f ∈ L2(Ω), then solution u ∈W 2

2 (Ω,w),weigth w exploding at the boundary (Babuska-Kondratiev spaces)

3. use global Sobolev reg.: If f ∈ L2(Ω), then solution u ∈W3/22 (Ω), use

counting argument:

#ηI ,p boundary wav.,diam(I ) ∼ 2−j ∼ 2j(d−1)



Besov regularity for p = 2 (ii)

Proof Idea:

extend u to Rn and take its wavelet decomposition – 3 parts

1. father wavelets (independent of regularity)2. interior and exterior wavelets ηI ,p with

dist(I , ∂Ω) & diam(I ) (1)

3. boundary wavelets ηI ,p; (1) doesn’t hold

handle 3 parts separately

1. no problem2. use weighted Sobolev reg.: If f ∈ L2(Ω), then solution u ∈W 2

2 (Ω,w),weigth w exploding at the boundary (Babuska-Kondratiev spaces)

3. use global Sobolev reg.: If f ∈ L2(Ω), then solution u ∈W3/22 (Ω), use

counting argument:

#ηI ,p boundary wav.,diam(I ) ∼ 2−j ∼ 2j(d−1)


Sobolev and local Holder regularity

Table of contents







Sobolev and local Holder regularity – Sobolev regularity of the p-Laplace

Sobolev regularity of the p-Laplace

Theorem (Ebmeyer 2001, 2002, Savare 1998)

Ω ⊂ Rd bounded polyhedral domain, d ≥ 2, 1 0 (2)

Furthermore

|∇u| ∈ Lq(Ω) for q <pd

d − 1

and

u ∈

W

3/2−εp (Ω), if 1 p.

Open question: Does (2) hold for general Lipschitz domains?C. Ebmeyer. Nonlinear elliptic problems with p-structure under mixed boundary valueconditions in polyhedral domains. Adv. Diff. Equ., 6:873–895, 2001.


Sobolev and local Holder regularity – Local Holder regularity of the p-Laplace equation

Local Holder regularity of the homogen. p-Laplace

Replacement for the local (weighted) Sobolev regularity (p = 2)

Theorem (Lewis 1983; Ural’ceva; Evans; DiBenedetto;. . .)

Ω ⊂ Rd bounded open set, d ≥ 2, 1 0 s.t. B(x0, 64r) ⊂ Ω

maxx∈B(x0,r)

|∇u(x)| ≤ C

( ∫−

B(x0,32r)|∇u|pdx

)1/p

≤ C · r−d/p,

maxx ,y∈B(x0,r)

|∇u(x)−∇u(y)| ≤ C · r−α( ∫−

B(x0,32r)|∇u|pdx

)1/p

|x − y |α.

⇒ local (weighted) Holder regularity for homogeneous p-Laplace

J. Lewis. Regularity of the derivatives of solutions to certain degenerate elliptic equations.

Indiana Univ. Math. J., 32(6):849–858, 1983.



Local Holder regularity of the homogen. p-Laplace

Replacement for the local (weighted) Sobolev regularity (p = 2)

Theorem (Lewis 1983; Ural’ceva; Evans; DiBenedetto;. . .)

Ω ⊂ Rd bounded open set, d ≥ 2, 1 0 s.t. B(x0, 64r) ⊂ Ω

maxx∈B(x0,r)

|∇u(x)| ≤ C

( ∫−

B(x0,32r)|∇u|pdx

)1/p

≤ C · r−d/p,

maxx ,y∈B(x0,r)

|∇u(x)−∇u(y)| ≤ C · r−α( ∫−

B(x0,32r)|∇u|pdx

)1/p

|x − y |α.

⇒ local (weighted) Holder regularity for homogeneous p-Laplace

J. Lewis. Regularity of the derivatives of solutions to certain degenerate elliptic equations.

Indiana Univ. Math. J., 32(6):849–858, 1983.



Local Holder regularity of the inhomog. p-Laplace

We can transfer the local Holder regularity from the homogeneouscase to the inhomogeneous p-Laplace equation:

Theorem (Kuusi,Mingione 2013; Diening,Kaplicky,Schwarzacher)

Ω,d,p as before. Let

α∗ = supα : Theorem of Lewis holds including the estimates.

Then for u with ∆pu = f ∈ C 1,β(α):

u is locally α-Holder continuous for α < min(α∗, 1/(p − 1)).

Analog estimates hold for local Holder-seminorm of u.

Problem: α∗ ∈ (0, 1] is unknown for d ≥ 3. (later: case d = 2)

T. Kuusi and G. Mingione. Guide to Nonlinear Potential Estimates. Bull. Math. Sci,4(1):1–82, 2014.



Local Holder regularity of the inhomog. p-Laplace

We can transfer the local Holder regularity from the homogeneouscase to the inhomogeneous p-Laplace equation:

Theorem (Kuusi,Mingione 2013; Diening,Kaplicky,Schwarzacher)

Ω,d,p as before. Let

α∗ = supα : Theorem of Lewis holds including the estimates.

Then for u with ∆pu = f ∈ C 1,β(α):

u is locally α-Holder continuous for α < min(α∗, 1/(p − 1)).

Analog estimates hold for local Holder-seminorm of u.

Problem: α∗ ∈ (0, 1] is unknown for d ≥ 3. (later: case d = 2)

T. Kuusi and G. Mingione. Guide to Nonlinear Potential Estimates. Bull. Math. Sci,4(1):1–82, 2014.



Locally weighted Holder spaces C `,αγ,loc(Ω)

∼ 1

∼ 1‖ ‖C 1,α ∼ 1

∼ 2−1

∼ 2−1∼ 2γ

∼ 2−k

2−k

2kγ

C `,αγ,loc(Ω). . . Holder space,

locally weighted, with

`. . . number of derivatives

α. . . Holder exponent ofderivatives of order `

γ. . . growth of Holder exp.with distance to ∂Ω



Locally weighted Holder spaces C `,αγ,loc(Ω) (ii)

Definition (Locally weighted Holder spaces)

K compact subset of Ω, δK distance to ∂Ω, K family of compactsubsets of Ω, g ∈ C `(Ω), set

|g |C `,α(K) :=∑|ν|=`

supx ,y∈K ,

x 6=y

|∂νg(x)− ∂νg(y)||x − y |α

,

|g |C 1,αγ,loc(K)

:= supK∈K

δγK |g |C `,α(K) <∞,

C `,αγ,loc(Ω;K) =

g ∈ C `(Ω) : |g |

C `,αγ,loc(K)<∞

.

K shall be the set of all B(x0, r) such that B(x0, 64r) ⊂ Ω.

This definition (` = 1) is perfectly adapted to Lewis’ Theorem.



Local Holder regularity of the p-Laplace

Although the optimal local Holder regularity of the solution of thep-Poisson is unknown (d ≥ 3), we can estimate γ by Lewis’ Theorem

maxx ,y∈B(x0,r)

|∇u(x)−∇u(y)| ≤ C · r−α( ∫−

B(x0,32r)|∇u|pdx

)1/p

|x − y |α

≤ C · r−α( ∫−

B(x0,32r)|∇u|qdx

)1/q

|x − y |α, p ≤ q.

≤ C · r−α−d/q · ‖∇u Lq(Ω)‖ · |x − y |α.

Hence, using the result of Ebmeyer

|∇u| ∈ Lq(Ω) for q <pd

d − 1,

we are allowed to choose

γ = α + (d − 1)/p + ε for all ε > 0.



The case d = 2: Holder regularity of the p-Poisson (i)

Theorem (Lindgren, Lindqvist 2013; (DDHSW 2014))

Ω ⊂ R2 bounded polygonal domain, 1 < p <∞, f ∈ L∞(Ω). If∆pu = f , u = 0 on ∂Ω, then u is locally α-Holder continuous for all

α <

1, if 1 < p ≤ 2,

1p−1 , if 2 < p <∞.

Furthermore, for the same α’s, it holds

u ∈ C 1,αγ,loc(Ω) for γ = α + 1/p + ε.

The regularity 1p−1 is a natural bound, take v(x) = |x |p/(p−1).

homogen. case: Iwaniec, Manfredi (1989) proved u ∈ C `,αloc (Ω) with

`+ α = 1 +1

6

(1 +

1

p − 1+

√1 +

14

p − 1+

1

(p − 1)2

)>max

(2,

p

p − 1

)Benjamin Scharf Besov regularity of solutions of the p-Laplace equation 18 of 23

Besov regularity of the p-Laplace equation

Table of contents







Besov regularity of the p-Laplace equation – From Bsp,p (Ω) and C

`,αγ,loc

(Ω) to Bστ,τ (Ω)

From B sp,p(Ω) and C `,α

γ,loc(Ω) to Bστ,τ(Ω)

Theorem (Dahlke, Diening, Hartmann, S., Weimar(DDHSW) ’14)

Ω ⊂ Rd bound. Lipschitz dom., d ≥ 2, s > 0, 1 < p <∞, α ∈ (0, 1],

σ∗ =

`+ α, if 0 < γ < `+α

d + 1p ,

dd−1

(`+ α + 1

p − γ), if `+α

d + 1p ≤ γ < `+ α + 1

p ,

then for all

0 < σ < min

σ∗,

d

d − 1s

and

1

τ=σ

d+

1

p

we have the continuous embedding

Bsp,p(Ω) ∩ C `,α

γ,loc(Ω) → Bστ,τ (Ω).

If γ not too bad and local Holder regularity `+ α is higher thanSobolev regularity s, Besov regularity σ is higher than Sobolev reg. !


Besov regularity of the p-Laplace equation – Besov regularity of the p-Laplace

The case d = 2: Besov regularity of the p-Poisson

1. By Ebmeyer’s result

u ∈

B

3/2−εp,p (Ω), if 1 < p ≤ 2,

B1+1/p−εp,p (Ω), if p ≥ 2,

2. Lindgren, Lindqvist:

u ∈ C 1,αγ,loc(Ω), γ = α + 1/p + ε, α <

1, if 1 < p ≤ 2,

1p−1 , if 2 < p <∞.

3. γ not too bad? α + 1p + ε = γ

?< `+α

d + 1p = 1+α

2 + 1p ? Yes, α < 1

4. General embedding theorem, 1τ = σ

d + 1p ,

u ∈ Bστ,τ (Ω) for all σ <

2, if 1 3/2, if 1 1 + 1

p if 2 

12 , if 1 2,

and γ not too large depending on d . This implies

p ∈ (pd ,∞) with pd →∞ for d →∞.

E. Lindgren and P. Lindqvist. Regularity of the p-poisson equation in the plane.arXiv:1311.6795v2, 2013.

T. Iwaniec and J. Manfredi. Regularity of p-harmonic functions on the plane. Rev. Mat.Iberoamericana, 5(1-2):119, 1989.



Open problems

d = 2, can one do better, in dependency of the angles of theboundary?

Is the chosen γ optimal? non-polyhedral domains measure Besov regularity in Lq for the p-Laplace (q 6= p)

bring the F s,rlocp,q (Ω) spaces into play...

work in progress. . .

Thank you for your attention

e-mail: [email protected]

web: http://www-m15.ma.tum.de/Allgemeines/BenjaminScharf


[email protected]

http://www-m15.ma.tum.de/Allgemeines/BenjaminScharf


Open problems

d = 2, can one do better, in dependency of the angles of theboundary?

Is the chosen γ optimal? non-polyhedral domains measure Besov regularity in Lq for the p-Laplace (q 6= p)

bring the F s,rlocp,q (Ω) spaces into play...

work in progress. . .

Thank you for your attention

e-mail: [email protected]

web: http://www-m15.ma.tum.de/Allgemeines/BenjaminScharf


[email protected]

http://www-m15.ma.tum.de/Allgemeines/BenjaminScharf

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Besov regularity of solutions of the p-Laplace …Besov regularity of solutions of the p-Laplace...

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