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UntitledREGULARITY OF A FOURTH ORDERNON-LINEAR PDE WITH CRITICAL EXPONENTSun-Yung A. Chang, Matthew J. Gursky and Paul C. Yangx0. Introdu tionOur aim in this paper is to study the regularity of minimizers for a ertain vari-ational problem dened either on a losed ompa t four-dimensional Riemannianmanifold or in a smooth bounded domain in R4 . The asso iated Euler-Lagrangeequation is fourth order and non-linear; moreover, the non-linearity is \ riti al" ina sense whi h we will soon des ribe.The original inspiration for this work was a problem in onformal geometry.However, stripped of its geometri signi an e it is also an intriguing PDE problemwhi h involves su h issues as the best onstant for imbeddings of Sobolev spa esinto Orli z lasses and weak ompa tness phenomena. To illustrate this, and inorder to provide motivation for the fun tional we will eventually be studying, let usbegin by sket hing some well known results on the \fun tional determinant" of theLapla ian on surfa es.Let (M2; g) be a losed, ompa t Riemann surfa e and g the Lapla e-Beltramioperator on M2. In surfa e theory, the simplest hanges of metri are onformal;that is, we multiply g by a smooth positive fun tion whi h we write as gw = e2wg.The fun tional determinant quanties the ee t of a onformal hange of metri on the spe trum of the Lapla ian. It does so by measuring the hange of the log-determinant, formally dened aslog detg = log 1 + log 2 + (0:1)where 0 < 1 2 are the non- zero eigenvalues of g. While (0.1) is not welldened a priori, there is a way of regularizing the denition due to Ray and Singer([RS). The details of this pro edure are beyond the s ope of this introdu tion, butfor our purposes the important result is the formula of Polyakov ([Po) whi h givesthe ratio of the log-determinants under a onformal hange of metri :log detgwdetg = 112 ZM2 jrwj2 + 2Kw dA (0:2)Resear h of the rst author is supported in part by NSF Grant DMS-9401465.Resear h of the se ond author is supported in part by NSF Grant DMS-9623048Resear h of the third author is supported in part by NSF Grant DMS-9300881.Typeset by AMS-TEX1
where K is the Gauss urvature of g.Sin e the eigenvalues of the Lapla ian s ale like the re ipro al of distan e squared,the log-determinant is not invariant under w 7! w + for 2 R. We thereforeintrodu e the related fun tionalS[w = 112 ZM2 jrwj2 + 2KwdA ZM2 KdA log ZM2 e2wdA (0:3)where R dA = (area M2)1 R dA. This amounts to normalizing the area of themetri gw in (0.2) to have the same area as (M2; g).The Euler equation for (0.3) isw + (Z KdA)e2w = K; (0:4)where w 2 W 1;2(M2). By the Moser-Trudinger inequality ([T, see also [Mo), ifw 2 W 1;2(M2) then ew 2 Lp(M2) for any p 1; thus a weak solution to (0.4) isautomati ally in W 2;p(M2) for any p 1. Full regularity of riti al points of (0.3)is then immediate.The geometri signi an e of (0.4) is the following. Suppose w 2 C1(M2)satises (0.4). If Kw denotes the Gauss urvature of the metri gw = e2wg, then bya well known identity w +Kwe2w = K: (0:5)Comparing (0.4) and (0.5) we see that gw has onstant Gauss urvature. Thisapproa h was used by Osgood, Phillips, and Sarnak ([OPS1, [OPS2) to give analternative proof of the lassi al uniformization theorem.The dimension of interest to us is four, and we shall see that in this ase thetheory is more ompli ated. The starting point is the Polyakov-type formula dueto Branson and Orsted ([BO, see also [BCY, [CY1), whi h holds for dierentialoperators satisfying ertain \naturality" and \ onformal" assumptions whi h arerather te hni al in nature but are satised, for example, by the onformal Lapla ianand the square of the Dira operator. Before writing it down, though, we have toestablish some notation.Let (M4; g) be a four-dimensional Riemannian manifold, and W; Ri , and Rdenote the Weyl, Ri i, and s alar urvature of g, respe tively. Assuming M4 isoriented, we also have the de ompositionW = W+W into the self-dual and anti-self-dual omponents of the Weyl tensor. We further dene the urvature invariantQ = 112R 3jRi j2 + R2: (0:6)The onformal transformation rule for Q involves a fourth order onformallyinvariant operator rst dened by Paneitz ([P): P = 2+ Æh 23Rg2Ri i d, whered is the exterior derivative and Æ is the divergen e, the formal adjoint of d. Ifgw = e2wg, then Qw = Q(gw) is related to Q = Q(g) byPw + 2Q = 2Qwe4w: (0:7)2
Of ourse, one should ompare (0.7) to the orresponding identity for the Gauss urvature in (0.5).With these denitions, the formula of Branson and Orsted is given bylog det Agwdet Ag = +1 I+[w + 1 I [w+ 2 II[w + 3 III[w (0:8)where the oeÆ ients ()i depend on the operator A andI[w = 4 Z jWj2w dv Z jWj2 dv log Z e4w dv (0:9)II[w = hPw;wi+ 4 Z Qw dv Z Q dv log Z e4w dv (0:10)III[w = 12 Y [w 13 Z (R)w dv (0:11)Y [w = Z ewew 2 dv 13 Z Rjrwj2 dv; (0:12)where dv denotes the volume form on (M4; g).A very general existen e theory for exteremals of (0.8) was developed by the rstand third authors in [CY1. They ([CY1, [CY2 and also the survey arti le [C) alsostudied the \sub-fun tional" II separately, be ause of its parallel with (0.3). Theirexisten e results along with present regularity work have resulted in some interestingappli ations of the fun tional determinant to four-dimensional onformal geometry.For example, pursuing the analogy between the quantity Q on a four-manifold andthe Gauss urvature of a surfa e, in [Gu a strong vanishing theorem was proved forthe rst Betti number of a ompa t four-manifold of positive s alar urvature withR Q dv > 0.Turning to regularity, the Euler equation for (0.8) has little in ommon withits two-dimensional ounterpart in (0.3). Indeed, the sub-fun tional III introdu esnon-linearities into the equation whi h do not appear in (0.4). To see this, let usintrodu e a general lass of fun tionals F : W 2;2(M4) ! R, of whi h (0.8) is aparti ular example.Let F [w = Z [(w)2 + (w + jrwj2)2dv+ Z [Dij(riw;rjw) + E(w w)dv (A1)where ; 2 R and E : R ! R, D 2 Sym2(T M4) satisfyjE(x)j a1 ea2jxjjDijvivj j a3jvj2 (A2)3
and jE0(x)j a1 ea2jxj; (A3)where a1, a2 and a3 are onstants. Note that by the Moser{Trudinger inequality([Mo, [T), w 2W 2;2 implies ew 2 Lp for any p > 1; however if E(x) has faster thanexponential growth at innity then E(w) may not be integrable, hen e somethinglike assumption (A2) is ne essary.The Euler equation of F (see se tion 1) is2w = 1(jr2wj2 (w)2 + Ri (rw;rw))+ 2wjrwj2 + 3r2w(rw;rw)+ 4Dijrirjw + 5ÆjDrjw+ 6(E0(w w) E0(w w)): (0:13)In (0.13), ÆD is the divergen e of D 2 Sym2(T M4), bars above a quantity denotesthe mean value, and i = i(; ) for 1 i 6 are onstants.>From the imbeddings W 2;2(M4) ,! W 1;4(M4) and W 2;2(M4) ,! eL2 , one seesthat the RHS of (0.13) is a priori only in L1(M4). Therefore, we annot apply abootstrap argument as we did when analyzing (0.4). While there are no third orderderivatives, the se ond and rst order terms appear at riti al powers, makingregularity a deli ate question. Our main result is the following:Main Theorem On a ompa t four-dimensional manifoldM4, let w 2W 2;2(M4)be a minimizer of the fun tional F in (A1), subje t to the onditions (A2) and (A3).Then w 2 C1(M4).As a orollary we have the following interior regularity result for R4 asmooth, bounded domain. Assume now that F is dened with respe t to the Eu- lidean metri , and let f; g 2 C0( ). Consider the variational probleminfw2A F [w (0:14)where A = fw 2 C1( ) : w = f; wn = g on g. ThenCorollary Suppose w 2 W 2;2( ) is a minimizer of (0.14), where F is subje tto the onditions (A2) and (A3). Then w 2 C1( ).The appropriate model for the Euler equation (0.13) is in many respe ts theharmoni map equation in dimension 2. Re all that if (M2; g) and (Nk; h) areRiemannian manifolds of dimension 2 and k respe tively, then a map u :M2 ! Nkis harmoni if it is a riti al point of the energyE(u) = ZM2 jduj2dA (0:15)where du : TM2 ! TNk is the dierential of u. If one imbeds Nk into someEu lidean spa e Rm , then the Euler equation of (0.14) isu = A(du; du) (0:16)4
where A is the se ond fundamental form of Nk Rm . Note that, in analogy with(0.13), for a weak solution u 2 W 1;2(M2; Nk) = fu : M2 ! Nk : R jduj2 < 1g of(0.15), the RHS is a priori only in L1. Again, lower derivatives appear at riti alpowers.For minimizing harmoni maps from surfa es, regularity was established by Mor-rey ([M). The argument pro eeded in two stages. First, Morrey proved Holder ontinuity of an energy minimizing solution u by verifying energy de ay of the formRB(r) jruj2 Cr. This is done by omparing the energy integral to that of aharmoni fun tion having the same boundary data as the solution. Similarly, weaim to prove Holder ontinuity of minimizing solutions w by verifying energy de ayof the form: D(r) = ZBr jr2wj2 + jrwj4 + w2 = O(r )for some > 0. This is done by omparing the energy of w to that of a biharmoni fun tion h having the same data as w up to the rst order derivatives. One diÆ ultythat does not appear in the harmoni map problem is the omparison of the Hessiangrowth with the growth of the Lapla ian. It is intuitively lear that the dieren eshould be lower order and is a boundary term. In this omparison argument werely on a formula ([CQ2, see also [CQ1) whi h displays the third order derivativesof the omparison fun tion on the boundary B(r) as a pseudodierential operatoron the boundary data alone.The se ond stage of our regularity argument is patterned after the regularityresult of S hoen ([S) and S hoen-Uhlenbe k ([SU) for Holder ontinuous harmoni maps. This is based on a de ay estimate:G(r) = 1r4 ZB(r)(rp(w)2 + jrwj2 + 1)dx for some 0 < p < 2. Su h a de ay estimate allows us to bootstrap the Holder boundto a C1;1 bound jr2wj1 :The full regularity of the solution is an easy onsequen e of this pointwise estimate.It is perhaps worthwhile to remark that in analogy with the situation of harmoni maps in two dimensions, one would expe t the regularity of all solutions of equation(0.13). For, if one rewrites the RHS of (0.13) intrinsi ally (that is, as urvatureterms of the metri gw = e2wg) then olle tively the important terms are the four-dimensional Gauss{Bonnet integrand. On the other hand, the urvature invariantsthat o ur in the Gauss{Bonnet integrand are a tually losed top degree forms, andhen e may be thought of as divergen e terms in determinant form. These are thekey features that enter into the regularity result of Helein ([H1, [H2, see also [Band [E) for harmoni maps of surfa es.In subsequent work of the rst and third authors with L. Wang, the regularityof weak biharmoni maps from ompa t four-manifolds is investigated. In somesense this represents a generalization of the aforementioned work of Helein ([H1,5
[H2) to higher dimensions. The analysis involved is somewhat dierent from thepresent paper, as one must take advantage of a ertain an ellation phenomenon inthe Euler equation.We on lude the introdu tion with an explanation of the organization of thepaper. In se tion 1 the Euler-Lagrange equation for the fun tional F in (A1) isderived. In se tion 2 we prove some estimates on the biharmoni extension in termsof the boundary data, and provide some te hni al lemmas. In se tion 3 we beginthe proof proper by establishing the Holder ontinuity of minimizers of F . Finally,in se tion 4 we show that Holder ontinuous solutions of (0.13) are regular.It is a pleasure to thank L. Wang for many onsultations on general ellipti theoryduring the preparation of this paper. We also thank Libin Mou for pointing out ate hni al error in an early version of this paper.
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1. The Euler equationWe begin with a simple rst variation al ulation of the fun tional F .Proposition 1.1. Let w 2W 2;2, ' 2W 2;2 \ C . ThenF 0w(') ddtF (w + t')t=0= Z 2(1 + 2)w' dv+ Z 'f4[jr2wj2 (w)2 + Ri (rw;rw) 42wjrwj2 82r2w(rw;rw) 2Dijrirjw 2ÆjDrjw+ E0(w w) E0(w w)g dv (1.1)where E0 denotes the derivative of the fun tion E : R ! R and E0(w w) =(vol M)1 R E(w w)dv.Corollary 1.2. If w 2W 2;2 is a riti al point of F that is, if F 0w(') = 0 for ea h' 2W 2;2 \ C { then w weakly satises2w = 1(jr2wj2 (w)2 + Ri (rw;rw))+ 2wjrwj2 + 3r2w(rw;rw)+ 4Dijrirjw + 5ÆjDrjw+ 6(E0(w w) E0(w w)) (1.2)for onstants i = i(; ).Proof of Proposition. 1.1 From (A1),ddt F (w + t')jt=0= ddtZ (w + t')2dv+ Z (w + t'+ jrwj2 + 2thr';rwi+ t2jr'j2)2 dv+ Z (Dijriwrjw + 2tDijri'rjw + t2Dijri'rj'+E(w w + t(' ')) dvt=0= Z 2(1 + 2)w' dv+ Z 2[2hr';rwiw +'jrwj2 dv+ Z [42hr';rwijrwj2 + 2Dijri'rjw + (' ')E0(w w) dv= I + II + III : (1.3)7
The term II evidently ontains third order derivatives in w if we were to inte-grate by parts. However, it is possible to take advantage of a ertain an ellationphenomenon:Lemma 1.3. If w 2W 2;2, ' 2W 2;2 \ C , thenZ (2hr';rwiw +'jrwj2)dv= Z 2'[jr2wj2 (w)2 + Ri (rw;rw)dv (1.4)Proof. If we an show that (1.4) holds when w 2 C1 then it follows by a standardmollifying argument that it holds for w 2W 2;2 as well.So suppose w 2 C1. ThenZ 2hr';rwiw dv= Z (2'hrw;r(w)i 2'(w)2)dv : (1.5)Also, by the standard Bo hner formula,Z 'jrwj2 dv = Z 'jrwj2 dv= Z (2'jr2wj2 + 2'Ri (rw;rw)+ 2'hrw;rwi dvadding the above identity to (1.5) we get (1.4), and the lemma follows. By (1.4), II = Z 2[2hr';rwiw +'jrwj2dv= Z 4'[jr2wj2 (w)2 +Ri (rw;rw)dv : (1.6)Finally, a simple integration by parts givesIII = Z '[42wjrwj2 82r2w(rw;rw) 2Dijrirjw 2ÆjDrjw+ E0(w w) E0(w w)dv : (1.7)Substituting (1.6) and (1.7) into (1.3) we get (1.1). 8
x2. Preliminary estimates for bi-harmoni fun tionsGiven the form of the fun tional F in (A1), it is not surprising that the proofof regularity relies in part on omparing a minimizer w of F to its biharmoni extension on a small ball. In this se tion we therefore olle t some general resultson biharmoni extensions.Let B(r) = fx 2 R4 : jxj < rg denote the ball of radius r in Eu lidean spa e, entered at the origin. We let 0 and r0 denote the eu lidean Lapla ian andgradient (so as not to onfuse them with the manifold Lapla ian and gradient).Also, n will denote the outward normal derivative on B(r); T and rT thetangential Lapla ian and gradient.Proposition 2.1. Suppose h satises8>><>>:20h = 0 in B(r)hn = on B(r)h = on B(r) (2.1)where kkLp(B(r)) K1 for some p > 3; (2.2)k kC(B(r)) K2: (2.3)Then for ea h x 2 B(r) and x0 = rxjxj ,jh(x) h(x0)j . K1jx x0j +K2jx x0j1 3p : (2.4)Remark. The proof of Proposition 2.1 is quite routine. We will write down theproof here for the sake of ompleteness and in the pre ise form we will quote laterin this arti le.Proof. The proof of Proposition 2.1 relies on some fairly detailed knowledge of theGreen's fun tion for the bi-Lapla ian. LetG(x; y) = log 1S(x; y) (1 S(x; y)); (2.5)S(x; y) = r2jx yj2jyj2jx yj2 ; (2.6)y = r2yjyj2 : (2.7)A dire t al ulation gives for x 2 B(r), y 2 B(r),G(x; y)jy2B(r) = 0 (2.8)9
nyG(x; y)jy2B(r) = 0 (2.9)(0)yG(x; y)jy2B(r) = 4r2 jxj2r 2 1jx yj4 (2.10)ny (0)yG(x; y)jy2B(r) = 8r2 jxj2r 3 1jx yj6 (2.11)(0)2yG(x; y) = Æx (2.12)where ny denotes the outward normal derivative with respe t to y, (0)y denotesthe Lapla ian with respe t to y, and Æx is the Dira measure at x.We then have the following reprodu ing result:Lemma 2.2. If h satises (2.1), then for x 2 B(r),h(x) = IB(r) ny (0)yG(x; y) (y)ds(y) IB(r)(0)yG(x; y)'(y)ds(y) : (2.13)To prove (2.13) one simply applies Green's theorem twi e; we will omit thedetails.The next lemma on erns the integral kernels appearing in (2.13).Lemma 2.3. Given x 2 B(r), let x = dist(x; B(r)) = r jxj. Then for ea h > 32 , IB(r) 1jx yj2 ds(y) C 123x : (2.14)Proof. . The proof of (2.14) is a version of the standard teles oping argument forPoisson kernels. In what follows, C will denote a onstant whose value may hangefrom line to line but whi h only depends on .Let y 2 B(r), x 2 B(r), and x0 = rxjxj (i.e., x0 is the point on B(r) whi h is losest to x). We have two ases to onsider.First, suppose r x r=2. Then learly jx yj r=2 x=2, so thatIB(r) 1jx yj2 ds(y) C2x IB(r) ds(y) C2x r3 C2x (2x)3 C 123x :10
Now let us suppose that x < r=2. In this ase it is not diÆ ult to see that thereis a onstant so that for all y 2 B(r),jx yj2 [jx x0j2 + jx0 yj2= 2x + jx0 yj2 : (2.15)Let A0 = fy 2 B(r) : jy x0j xgAk = fy 2 B(r) : 2k1x < jy x0j 2kxg :Then by (2.15), y0 2 A0 ) jx yj2 C2x ;while learly IA0 ds(y) 3x :Hen e IA0 1jx yj2 ds(y) C 123x : (2.16)It also follows from (2.15) thaty 2 Ak ) jx yj2 C(2k1x)2 ;while IAk ds(y) (2kx)3 :Hen e IAk 1jx yj2 ds(y) C 1(2k1x)2 IAk ds(y) C(2kx)3(2k1x)2= C 123x (232)k : (2.17)Combining (2.16) and (2.17) we haveIB(r) 1jx yj2 ds(y) = 1Xk=0IAk 1jx yj2 ds(y) C 123x "1 + 1Xk=1(232)k# :Sin e we are assuming that 3 2 < 0, the series above onverges and we on lude(2.14). 11
Proof of Proposition 2.1 If we take h 1, 0, 1 in (2.13), we get1 IB(r) ny (0)yG(x; y)ds(y)for ea h x 2 B(r).Fix x 2 B(r), denote x0 = rxjxj , x = jx x0j = r jxj. Thenh(x) h(x0) = I + IIwhere I = IB(r) ny (0)yG(x; y)( (y) (x0))ds(y)II = IB(r)(0)yG(x; y)(y)ds(y) :By (2.1) and our assumption (2.3), we havejIj r2 jxj2r 3 IB(r) K2jy x0jjx yj6 ds(y). K23x IB(r) 1jx yj6where in the last line we have used the fa t that jxyj jxx0j for all y 2 B(r).Applying Lemma 2.3, we getjIj K23x C3(6)x . K2x :Applying (2.10), we havejIIj 4r2 jxj2r 2 IB(r) j(y)jjx yj4 ds(y). 2xkkLp(B(r)) IB(r) ds(y)jx yj4p0! 1p 0where 1p + 1p0 = 1. Sin e p > 3, p0 > 34 , so by Lemma 2.3, we havejIIj . K12x 34p0x 1p0 = K11 3px :Combining the estimates in I and II, we getjh(x) h(x0)j . K2x +K11 3px ;and this nishes the proof of Proposition 2.1. The next result appears in [CQ2, and will be used throughout the regularityproof in se tion 3 below. 12
Proposition 2.4 ( [CQ2, Lemma 3.3). Suppose h satises (2.1). Thenh0hjB(r) = 2P3 + 2T 2T (2.18)0hjB(r) = 2T 2f(T + 1)1=2 + 1g (2.19)where P3 = (T )(T + 1)1=2 : (2.20)The identities are understood to hold whenever the RHS of (2.18) and (2.19)exist.We on lude with two te hni al results whi h will be used in se tion 3.Proposition 2.5. Suppose h satises (2.1) with8<: ' = wn = w : (2.21)ThenIB(r) jr20hj2ds C "IB(r)(jr20wj2 + r2jr0wj2)ds+ r1 ZB(r)(0h)2dx#(2.22)for some onstant C whi h does not depend on h or w.Proof. We havejr20hj2 . 2hn22 + jrT hn j2 + jrTrThj2 + r2jr0hj2 :Sin e w = h and hn = wn on B(r), this be omesjr20hj2 . 2hn22 + jrT wn j2+ jrTrThj2 + r2jr0hj2 2hn22 + jr20wj2 + jrTrThj2+ r2jr0hj2 :The integrated Bo hner identity on B(r) givesIB(r) jrTrThj2ds . IB(r)(Th)2ds13
so that IB(r)jr20hj2ds . IB(r) 2hn22 ds+ IB(r) jr20wj2 + (Th)2 + r2jr0wj2 ds : (2.23)We now laimIB(r) 2hn22 ds . IB(r) jr20wj2ds+ IB(r) r2jr0wj2ds; (2.24)whi h more or less follows from the proof of [CQ2, Lemma 3.3, but we will give abrief sket h of the proof. Suppose the expansion in spheri al harmoni s fYkg of wand wn on B(r) are given by 8>>><>>>: w =Xk akYkwn =Xk bkYk : (2.25)Using [CQ2, Lemma 3.3 we an write the expansion of h in B(r), and dierentiationyields 2hn2 jB(r) =Xk fk(k + 2)ak (2k + 1)bkgYk :Therefore, IB(r) 2hn22 ds Xk fk2(k + 2)2a2k + (2k + 1)2b2kg : (2.26)Using (2.25) we also haveIB(r)(Tw)2ds =Xk k2(k + 2)2a2k ;IB(r) (T + 1)1=2 wn 2 =Xk (k + 1)2b2k ;whi h when ompared to (2.26) givesIB(r) 2hn22 ds . IB(r)((Tw)2 + (T + 1)1=2wn 2) ds. IB(r) jr20wj2 + r2jr0wj2 ds :14
>From (2.23) and (2.24) we getIB(r)jr20hj2ds IB(r) jr20wj2ds+ IB(r)(Th)2ds+ r2 IB(r) jr0wj2ds : (2.27)Writing 0jB(r) in terms of T we see that0h = 2hn2 + 3r hn +Th= 2hn2 + 3r wn +Th)IB(r)(Th)2ds. IB(r) "2hn22 + (0h)2 + r2wn2# ds. IB(r) jr20wj2 + (0h)2 + r2jr0wj2 ds : (2.28)By (2.19),0hjB(r) = 2Tw + 2f(T + 1)1=2 + 1gwn)IB(r)(0h)2ds . IB(r) (Tw)2 + jr20wj2 + r2jr0wj2 ds. IB(r) jr20wj2 + r2jr0wj2 ds :Substituting this into (2.28) givesIB(r)(Th)2ds. IB(r) jr20wj2 + r2jr0wj2 ds : (2.29)Substituting (2.29) into (2.27) we getIB(r)jr20hj2ds IB(r) jr20wj2 + r2jr0wj2 ds : (2.30)Substituting (2.30) into (2.27) we arrive at (2.22). 15
Proposition 2.6. For f 2W 2;2(B(r)) \W 2;2(B(r)),ZB(r) jr20f j2dx . ZB(r)((0f)2 + r2jr0f j2)dx+ r IB(r) jr20f j2 + r2jr0f j2 ds (2.31)Proof. We will prove that (2.31) holds for f 2 C1(B(r)); the general result followsfrom a standard limiting argument.To begin, 120(jr0f j2) = jr20f j2 + hr0(0f) ;r0fi : (2.32)Integrating by parts givesZB(r)hr0(0f) ;r0fidx = ZB(r)(0f)2dx+ IB(r)0f fn : (2.33)Combining (2.32) and (2.33), we getZB(r) jr20f j2dx = ZB(r)(0f)2dx+ I + II (2.34)where I = 12 ZB(r)0(jr0f j)2dx;II = IB(r)0f fn dsTo estimate I, we haveI = 12 IB(r) n(jr0f j2)ds. IB(r) jr20f jjr0f jds. r IB(r) jr20f j2 + r2jr0f j2 ds (2.35)We an estimate II asjIIj . IB(r)(0f)2 ds!1=2 IB(r) fn2 ds!1=2 . IB(r) jr20f j2 ds!1=2 IB(r) jr0f j2 ds!1=2. r IB(r) jr20f j2ds+ r1 IB(r) jr0f j2ds: (2.36)16
Combining (2.34)-(2.36) we get (2.31).As an appendix to se tion 2, we will list here a number of elementary Sobolevinequalities whi h we will use throughout the paper. For simpli ity, we will list allof these inequalities for a ball B(r) in R4 .(S1) If f 2W 1;2(B(r)) then ZB(r) jf j4dx!1=2 . ZB(r) jr0f j2dx+ 1r2 ZB(r) jf j2dx(S2) If f 2W 1;2(B(r)) then ZB(r) jf j4dx!1=2 . ZB(r) jr0f j2dx+ 1r IB(r) jf j2 ds(P1) If f 2W 1;20 (B(r)) thenZB(r) jf j2 dx . r2 ZB(r) jr0f j2 dx(P2) If f 2W 2;20 (B(r)) thenZB(r) jr0f j2 dx . r2 ZB(r)(0f)2 dx(P3) If f 2W 1;2(B(r)) thenIB(r)(f f)2 ds . r2 IB(r) jrT0 f j2 dswhere f = 1jB(r)j HB(r) f ds(P4) If f 2W 2;2(B(r)) thenIB(r) jrT0 f j2 ds . r2 IBr (T0 f)2 ds : 17
x3. Holder Continuity of MinimizersIn this se tion we derive a preliminary regularity result for minimizers of (1.1);namely, we show that they must be Holder ontinuous. Sin e this is purely a lo alresult, it will be easier if we work in an open set U M whi h admits oordinates.So let fxig be normal oordinates entered at P 2 U . Thengij = Æij + O(jxj2)kgij = O(jxj) :In parti ular, U admits a Eu lidean metri g0 with orresponding volume form,Lapla ian, and gradient given by g0 = Æijdx = dx1 dx40 = 4Xi=1 2i(r0')i = i' :We let r denote the Eu lidean distan er2 = 4Xi=1(xi)2and denote Eu lidean balls entered at p byB(s) = fQ 2M : r(Q) < sg :The fa t that our oordinates are normal allows us ompute quantities in the Eu- lidean metri with a well- ontrolled error that depends on r = jxj. For example,dv = (1 + O(r2))dxjrwj2 = (1 + O(r2))jr0wj2(w) = gij(ijw + kijkw) (3.1)= (1 + O(r2))(0w) +O(r2)jr20wj+ O(r)jr0wjjr2wj2 = (1 + O(r2))jr20wj2 + O(r2)jr0wj2To prove Holder ontinuity, we will use a generalized form of Morrey's Lemma(see [GT, Theorem 7.19) whi h says that if there are onstants K, > 0 su h thatour minimizer w satises ZB(r) jr0wjdx Kr3+ (3.2)18
for any ball B(r), then kwkC C( ;K). We will a tually prove thatZB(r) jr0wj4dx Kr4 (3.3)whi h implies (3.2) via Holder's inequality.To this end, deneD(r) = ZB(r)[jr20wj2 + jr0wj4 + w2dx; (3.4)E(r) = D(r) + T (r);where T (r) = ZB(r) jxj2jr0wj2dx: (3.5)Let us begin with some preliminary remarks whi h explain when T (r) makes sense.First dene T(r) = ZB(r)B() jxj2jr0wj2dx:Then T(r) is well dened for > 0. Writing T(r) in polar oordinates we haveT(r) = Z r t2 IB(t) jr0wj2ds! dt:Integrating by parts we getT(r) = r1 IB(r) jr0wj2ds+ 1 IB() jr0wj2ds+ Z r t1 ddt IB(t) jr0wj2ds! dt : (3.6)Note that ddt IB(t) jr0wj2ds! = ddt IB(1) jr0w(t; )j2t3dwhere d is the surfa e measure on B(1). Thusddt IB(t) jr0wj2ds! = 3t1 IB(1) jr0w(t; )j2t3d+ IB(1) ddt jr0w(t; )j2 t3d:19
Substituting this ba k into (3.6) we getT(r) = r1 IB(r) jr0wj2ds+ 1 IB() jr0wj2ds+ 3T(r) + Z r t1 IB(1) ddt jr0w(t; )j2 t3ddt:Hen e,2T(r) r1 IB(r) jr0wj2ds+ 2 ZB(r)B() jxj1jr20wjjr0wjdx r1 IB(r) jr0wj2ds+ 2 ZB(r)B() jxj2jr0wj2dx!1=2 ZB(r) jr20wj2dx!1=2 r1 IB(r) jr0wj2ds+ T(r) + ZB(r) jr20wj2dx:Therefore, T(r) r1 IB(r) jr0wj2ds+ ZB(r) jr20wj2dx:Taking the limit as ! 0 we have provedLemma 3.1. If HB(r) jr0wj2ds <1, then T (r) <1 andT (r) r1 IB(r) jr0wj2ds+ ZB(r) jr20wj2dx: (3.7)Remark. Inequality (3.7) is a fourth order version of inequality (1.10) of [GL,whi h is related to the \Heisenberg un ertainty prin iple" (see [We).As w 2 W 2;2 \W 1;4, there is a onstant C su h thatD(r) C (3.8)for any r > 0. Moreover, D(r) is a monotone fun tion hen e dierentiable almosteverywhere. The following proposition is the main te hni al result of this se tion:Proposition 3.2. There is a onstant A = A( C) su h that for all r > 0 suÆ ientlysmall satisfying rD0(r) 4 C (3.9)one has E(r) ArE0(r) + Ar (3.10)for some 2 (0; 2.Before we present the proof of Proposition 3.2, let us see how the estimate (3.3)is a onsequen e. First, a lemma: 20
Lemma 3.3. For all r suÆ iently small, E(r) is bounded andE(r) CrE0(r) + Cr (3.11)holds for a onstant C depending on the onstant C and the onstant A in the on lusion of Proposition 3.2.Proof. It is important to note that given r0 > 0 small enough, there always existsan r 2 [r0=2; r0 whi h satises (3.9). For, given 2 (0; 1), there is a measurableset (r0=2; r0) su h that rD0(r) 21D(r0) (3.12)for all r 2 , and jj = (1 ) r02 (Proof: redu tio ad absurdum.) If we take = 12 , it follows that there is an r 2 1=2 withrD0(r) 4D(r0) 4 C :To prove Lemma 3.3, we will rst prove that for r suÆ iently small, E(r) is bounded.Suppose r > 0 is given. Choose r1 2 [r; 2r su h thatr1D0(r1) 32 inf2[r;2rD0() 32 rD0(r) : (3.13)Note that by taking = 34 we may on lude from (3.12) thatinf2[r;2rD0() 83 D(2r) :Hen e r1D0(r1) 32 83D(2r) = 4D(2r) 4 C :So by Proposition 3.2, E(r1) Ar1E0(r1) +Ar1 : (3.14)We now observe thatr1E0(r1) = r1D0(r1) + r1T 0(r1)= r1D0(r1) + r11 IB(r1) jr0wj2ds. r1D0(r1) + r1 IB(r1) jr0wj4ds!1=2. r1D0(r1) + (r1D0(r1))1=2. 4 C + 4 C1=2 :21
Thus we on lude from (3.14) that E(r) E(r1) is bounded. Consequently wemay repeat the arguments in (3.12) and (3.13) above for E(r) instead of D(r) and on lude from (3.14) thatE(r) E(r1) Cr1E0(r1) + Cr1 (By (3.14)) 32CrE0(r) + Cr1 (By (3.13)) 32CrE0(r) + C(2r) 2CrE0(r) + 4Crfor some onstant C = C( C;A). Sin e (3.11) holds for all r suÆ iently small, we an integrate the dierentialinequality to on lude E(r) Kr4 for some K = K( C), = ( C). This in turn implies (3.3).Let us now turn to the proof of Proposition 3.2.Proof. Our rst step is to onstru t a ompetitor to w for the fun tional (A1). Leth denote the biharmoni extension of w to B(r):8>><>>:20h = 0 in B(r)hn = wn on B(r)h = w on B(r) (3.15)here n denotes the (Eu lidean) outward normal derivative. We extend h to all ofM by letting h = w outside of B. Note that the assumption (3.9) ensures that theboundary data is well dened.It follows that h 2 W 2;2(M); in fa t, there is a variational hara terization of hwhi h is given by ZB(r)(0h)2dx ZB(r)(0')2dx (3.16)for all ' 2 W 2;2(B(r)) satisfying the same boundary onditions as h. From (3.16)we see that khk2;2 C(kwk2;2) : (3.17)The following lemma will provide the basi estimate that will allow us to proveProposition 3.2.Lemma 3.4.ZB(r)[(0w)2 + jr0wj4dx . ZB(r)[(0h)2 + jr0hj4dx+ r : (3.18)22
Proof. Sin e w is extremal, F [w F [h : (3.19)Given M , let us deneF 1 (') = Z [(')2 + ('+ jr'j2)2dv ;F 2 (') = Z [E(' ') +Dijri'rj'dv :Then F (') = F 1M (') + F 2M (') :Let B denote B(r). From (3.19) we haveF (w) = F 1B(w) + F 2B(w) + F 1MnB(w) + F 2MnB(w) F (h) = F 1B(h) + F 2B(h) + F 1MnB(h) + F 2MnB(h))F 1B(w) F 1B(h) + Z(w; h)where Z(w; h) = F 2B(h) F 2B(w)be ause h = w on MnB.Using the inequality 2xy (2 + 12)x2 22222+1y2 we haveF 1B(w) = ZB[(1 + 2)(w)2 + 2wjrwj2 + 2jrwj4dv ZB 12(w)2 + 21 + 22 jrwj4 dv& ZB[(w)2 + jrwj4dv (3.20)Likewise, F 1B(h) ZB [(1 + 22)(jh)j2 + 22jrhj4dv. ZB [(h)j2 + jrhj4dv : (3.21)23
To estimate the term Z(w; h) we use the assumption (A2) on E and (Dij) to getjZ(w; h)j ZB jE(w w)E(h h) +DijriwDjw Dijrihrjhjdv ZB [a1 exp a2jw wj+ a1 exp a2jh hj+ a3(jrwj2 + jrhj2)dv. "ZB e2a2jw wjdv1=2 + ZB e2a2jhhjdv1=2# jBj1=2+ hkrwk2L4(B) + krhk2L4(B)i jBj1=2where jBj = RB dv = RB(1 + O(r2))dx . r4. Sin e w, h 2W 2;2,ZB e2a2jw wjdv + ZB e2a2jhhjdv C ;hen e jZ(w; h)j . r (3.22)with = 2.Combining (3.20) - (3.23) we getZB [(w)2 + jrwj4dv . ZB [jhj2 + jrhj4dV + r :Using the estimates (3.1) we arrive at (3.18). We now wish to show that the estimates for h developed in se tion 2, along withthe boundary data of h, allow us to ompare the right hand side of (3.18) to theright hand side of (3.10).To begin, let us onsider the integralI = ZB(0h)2dx : (3.23)Using (3.15) and integrating by parts twi e we have0 = ZB h20hdx = ZB(0h)2dx+ IB h n(0h)ds IB 0hhnds (3.24)where H ds denotes boundary integrals and ds is the indu ed (Eu lidean) surfa emeasure. Rewriting (3.24), I = I1 + I2 ; (3.25)24
I1 = IB h n (0h) ds ; (3.26)I2 = IB 0h hn ds : (3.27)Let T and rT denote the Eu lidean tangential Lapla ian and gradient on B.By Proposition 2.4 n0h = 2P3w 2Tw + 2T wn (3.28)where P3 is the pseudo-dierential operatorP3w = T (T + 1)1=2w : (3.29)A word about the sense in whi h the third derivatives in (3.28) exist: For thegiven w 2 W 2;2(M4) satisfying the Euler equation (1.2), observe that 2w 2L1(M4). Therefore jr3wj 2 Lp(M4) for any p < 4=3, and by hoosing the ballB(r) suitably (whi h is allowed by the argument in Lemma 2.2) we may assumejr3wj 2 Lp(B). In parti ular, all the terms on the RHS of (3.28) exist in anLp(B)-sense for p < 4=3, whi h in turn implies (3.28) and also justies the inte-gration in (3.26).In any ase we haveI1 = IB 2wP3w Tw +T wn ds= Ia1 + Ib1 + I 1 :Using (3.29) and integrating by parts,Ia1 = IB 2wP3w ds= 2 IB (T + 1)1=2wTw ds= 2 IB[(T + 1)1=2w 1Tw ds ;where 1 = 1jBj IB(T + 1)1=2w ds ;jBj = IB ds :By the S hwarz inequalityjIa1 j 2IB h(T + 1)1=2w 1i2 ds1=2IB(Tw)2ds1=225
and by the Poin aire inequality (P3)IB[(T + 1)1=2w 12ds . r2 IB jrT (T + 1)1=2wj2ds ;so that jIa1 j . rIB jrT (T + 1)1=2wj2ds1=2IB(Tw)2ds1=2. r I jrTrTwj2ds. r IB jr20wj2ds+ r1 IB jr0wj2ds : (3.30)Using a similar argument, Ib1 = 2 IB wTw ds= 2 IB(w 2)Tw dswhere 2 = 1jBj IB w ds :Then using the Poin aire inequality (P4) we havejIb1j 2IB(w 2)2ds1=2I (Tw)2ds1=2. r4 IB(Tw)2ds1=2IB(Tw)2ds1=2. r2 IB jr20wj2ds+ IB jr0wj2ds : (3.31)Finally, I 1 = 2 IB wT wn ds= 2 IB Twwn dsHen e, jI 1j 2IB(Tw)2 ds1=2IB jr0wj2ds1=2. IB jr20wj2 ds+ r2 IB jr0wj2ds1=2 IB jr0wj2ds1=2. r IB jr20wj2 ds+ r1 IB jr0wj2ds (3.32)26
Combining (3.30) - (3.32) we on ludejI1j . r IB jr20wj2 ds+ r1 IB jr0wj2ds : (3.33)To estimate I2, we appeal on e more to Proposition 2.4 and write0h = 2Tw 2 h(T + 1)1=2 + 1i wnso that I2 = IB 0hhn ds= IBf2Tw 2[(T + 1)1=2 + 1wn g wn ds= Ia2 + Ib2 :Noti e jIa2 j = jI 1j . r IB jr20wj2 ds+ r1 IB jr0wj2ds : (3.34)Also, Ib2 = IB 2 h(T + 1)1=2 + 1i wn wn dsHen e, jIb2j . IB jr20wj+ r1jr0wj jr0wjds. r IB jr20wj2 ds+ r1 IB jr0wj2 ds: (3.35)Therefore, jI2j jIa2 j+ jIb2j. r IB jr20wj2 ds+ r1 IB jr0wj2 ds: (3.36)Combining (3.33) and (3.36) we getI . r IB jr20wj2 ds+ r1 IB jr0wj2 ds: (3.37)The next term to estimate in (3.18) isII = ZB jr0hj4 dx : (3.38)27
Applying the Sobolev inequality (S2) to the fun tion jr0hj and using the fa t thatjr0jr0hjj2 jr20hj2 we getZB jr0hj4 dx1=2 . ZB jr20hj2 dx+ r1 IB jr0hj2 ds: (3.39)Considering the se ond term on the RHS of (3.39) we use the de omposition ofthe gradient into tangential and normal omponents to getjr0hj2 = jrThj2 + hn2= jr0wj2:Therefore, ZB jr0hj4dx1=2 . ZB jr20hj2dx+ r1 IB jr0wj2 ds: (3.40)By the assumption (3.9) rD0(r) 4 CTherefore, r IB jr0wj4 4 C:Hen e, r1 IB jr0wj2ds r3=2r IB jr0wj4ds1=2IB ds1=2 C:It then follows from (3.8) and (3.17) thatZB jr0hj4dx1=2 . 1: (3.41)>From (3.40) and (3.41) we on ludeZB jr0hj4dx . ZB jr20hj2dx+ r1 IB jr0wj2 ds: (3.42)Appealing to the te hni al result Proposition 2.6 allows us to write the interiorHessian integral in (3.42) as an interior Lapla ian integral modulo a boundary term:ZB jr0hj4 dx . ZB(0h)2 dx+ r1 IB jr0wj2 ds+ r IB jr20hj2 ds : (3.43)28
Next we appeal to Proposition 2.5, whi h allows us to bound the last term on theright hand side of (3.43):ZB jr0hj4 dx . ZB(0h)2 dx+ r IB jr20wj2 ds+ r1 IB jr0wj2 ds :Finally, by (3.37) we on ludeII = ZB jr0hj4 dx . r IB jr20wj2 ds+ r1 IB jr0wj2 ds : (3.44)Combining (3.37), (3.44), and (3.18) we haveZB[(0w)2 + jr0wj4 dx . r IB jr20wj2 + r1 IB jr0wj2 ds+ r: (3.45)Re all that D(r) = ZB [jr20wj2 + jr0wj4 + w2 dx :Using the Sobolev inequality (S2),ZB w2dx jBj1=2ZB w4dx1=2. r2ZB jr0wj2dx+ r1 IB w2 ds. r2 ZB jr0wj4dx+ r IB w2ds+ r6. ZB jr0wj4dx+ r IB w2 ds+ r :Therefore, D(r) . ZB [jr20wj2 + jr0wj4dx+ r IB w2ds+ r : (3.46)Now, E(r) = D(r) + T (r);so by Lemma 3.1,E(r) . ZB[jr20wj2 + jr0wj4dx+ r IB [r2jr0wj2 + w2ds+ r:Taking f = w in Proposition 2.6,E(r) . IB[j0wj2 + jr0wj4dx+ r IB[jr20wj2 + r2jr0wj2 + w2ds+ r:And therefore by (3.45)E(r) . r IB[jr20wj2 + r2jr0wj2 + w2ds. rE0(r) + r ;whi h ompletes the proof of Proposition 3.2. 29
x4. Classi al regularity from Holder ontinuity of minimizersIn this se tion we omplete the proof of our regularity result by showing thatweak solutions to the Euler equation (0.4) whi h are Holder ontinuous are alreadyregular.As in Se tion 3, our estimates will be lo al. It will therefore simplify mattersif we work in a neighborhood of a point P 2 M whi h admits normal oordinates entered at P .Lemma 4.1. Suppose w 2 W 2;2 \ C is a riti al point of F . Then for r > 0suÆ iently small, given ' 2W 2;2 \ C 0 (B(r)) we haveZB(r)0w0'dx C(r + supB(r) j'j) ZB(r)[jr20wj2 + jr0wj4 + 1dx+ Cr(1 + supB(r) j'j) ZB(r)[1 + jr20'j2dx : (4.1)Proof. Suppose ' 2W 2;2 \ C 0 (B(r)). If F 0w(') = 0 then by (1.1)Z w' dv . Z j'j[jr2wj2 + jrwj4+ 1 + jE0(w w) E0(w w)jdv :By assumption (A3), jE0(w w)j a1exp a2jw wj. Sin e w 2 C we on ludeZ w' dv . Z j'j[jr2wj2 + jrwj4 + 1dv : (4.2)Using (3.1) and the fa t that supp ' B(r) we haveZ j'j[jr2wj2 + jrwj4 + 1dv= Z j'j[(1 +O(r2)jr20wj2 +O(r2)jr0wj2+ (1 + O(r2))jr0wj4 + 1(1 + O(r2))dx C supB(r) j'j ZB(r)[jr20wj2 + jr0wj4 + 1dx (4.3)Also by (3.1),Z w' dv = Z [0w + O(r2)jr20wj+ O(r)jr0wj [0'+ O(r2)jr20'j+ O(r)jr0'j(1 + O(r2))dx30
) Z 0w0'dx = Z w ' dv+ O(r) Z [jr20wjjr20'j+ jr20wjjr0'j+ jr0wjjr20'jdx) Z 0w0' dx Z w ' dv+ Cr Zsupp'[jr20wj2 + jr20'j2 + jr0wj2 + jr0'j2dx :Sin e supp ' B(r), Z jr0'j2dx = Z '0' dx supB(r) j'j ZB(r) j0'j dx C supB(r) j'j ZB(r)[1 + jr20'j2dx :Therefore, Z 0w0' dx Z w' dV + Cr ZB(r)[jr20wj2 + jr0wj4 + 1dx+ Cr(1 + supB(r) j'j) Z [1 + jr20'j2dx : (4.4)Combining (4.2), (4.3), and (4.4) we get (4.1). >From now on let us adopt the onvention that RB(s)dx = (vol B(s))1 RB(s) dx.Dene G(r) = ZB(r) [rp(0w)2 + jr0wj2 + 1dx ; (4.5)where 0 < p < 2 will be spe ied later. The following de ay estimate for G(r) willbe the main te hni al result of this se tion.Proposition 4.2. Suppose w 2W 2;2 is riti al for F and thatZB(r)[jr20wj2 + jr0wj4 + w2dx Kr4 (4.6)31
for some onstants K, > 0 and all r > 0 suÆ iently small if < 1. Then withp = 2 =4, G(r) C(K; ) : (4.7)Remark. If w 2 W 2;2 is a minimizer of F , then (4.6) follows from Lemma 3.2.Therefore, by Proposition 4.2 any minimizer of F satisesjr0wj C : (4.8)At the on lusion of the proof of Proposition 4.2 we will use (4.8) to argue thatw 2 C1.Proof. First, note that by (4.6) and our observations in x3 it follows that w 2 C .Fix r > 0 small enough so that B(r) is well dened and let h denotes thebiharmoni extension of w as in (3.15). From Proposition 2.1 we see that h 2C (B(r)) as well. If we extend h to be equal to w outside of B(r) it then followsthat ' = w h satises ' 2W 2;2 \ C 0 (B(r)).Our estimate of G(r) begins with an estimate ofZB(r=2)(0w)2dx = ZB(r=2)0w(0w 0h) dx+ ZB(r=2)0w 0h dx ZB(r=2)0w 0(w h) dx+ ZB(r=2)(0w)2dx1=2ZB(r=2)(0h)2dx1=2 ZB(r=2)0w0(w h)dx+ 12 ZB(r=2)(0w)2dx+ 12 ZB(r=2)(0h)2dx) ZB(r=2)(0w)2dx ZB(r=2)(0h)2dx+ 2 ZB(r=2)0w 0(w h)dx : (4.9)Examining the se ond term on the RHS of (4.9),ZB(r=2)0w0(w h)dx ZB(r=2)(0w)2dx1=2ZB(r=2) [0(w h)2dx1=2 C ZB(r)(0w)2dx1=2ZB(r) [0(w h)2dx1=2 (4.10)32
Noti e that ZB(r) [0(w h)2dx= ZB(r) [(0w)2 20w0h+ (0h)2dx= ZB(r) [20w 0'+ (0h)2 (0w)2dx : (4.11)Re all the variational hara terization of h as in (3.16), whi h in parti ular impliesZB(r)(0h)2dx ZB(r)(0w)2dx : (4.12)Therefore by (4.11), (4.1) we haveZB(r) [0(w h)2dx 2 ZB(r)0w 0' dx C r + supB(r) j'j!ZB(r) [jr20wj2 + jr0wj4 + 1dx+ Cr 1 + supB(r) j'j!ZB(r) [1 + jr20'j2dx : (4.13)Sin e ' 2 C , supB(r) j'j Cr . To simplify matters, let us assume < 1, so that r + supB(r) j'j! Cr : (4.14)To estimate the Hessian term in (4.13), we again appeal to the variational har-a terization of h in (3.13):ZB(r) jr20hj2dx ZB(r) jr20wj2dx :Therefore, ZB(r) jr20'j2dx = ZB(r) jr20(w h)j2dx 2 ZB(r)(jr20wj2 + jr20hj2)dx 4 ZB(r) jr20wj2dx : (4.15)33
Combining (4.13) - (4.15) we on ludeZB(r) [0(w h)2dx Cr ZB(r) [jr20wj2 + jr0wj4 + 1dx : (4.16)Therefore, by (4.9) and (4.16),2 ZB(r=2)0w0(w h)dx Cr =2ZB(r)(0w)2dx1=2ZB(r) [r20wj2 + jr0wj4 + 1dx1=2 Cr =2 ZB(r) [jr20wj2 + jr0wj4 + 1dx : (4.17)The next step requires areful estimation of the RHS of (4.17). To this end, we usethe Sobolev inequality (S1) with f = jr0wj:ZB(r) jr0wj4dx1=2 . r2 ZB(r) jr0jr0wjj2dx+ ZB(r) jr0wj2dx. r2 ZB(r) jr20wj2dx+ ZB(r) jr0wj2dx : (4.18)By (4.6) the RHS of (4.18) an be estimated as follows:r2 ZB(r) jr20wj2dx+ ZB(r) jr0wj2dx. r2 "ZB(r) jr20wj2dx+ r2 ZB(r) jr0wj2dx#. r2 "Kr4 + r2 Z jr0wj4dx1=2 jB(r)j1=2#. r2[Kr4 + CK1=2r2 . r2 2 :Therefore ZB(r) jr0wj4dx . r2 ZB(r) jr20wj2dx+ ZB(r) jr0wj2dx2 r2 2 r2 ZB(r) jr20wj2dx+ ZB(r) jr0wj2dx r2 ZB(r) jr20wj2dx+ r2 2 ZB(r) jr0wj2dx : (4.19)34
Combining (4.17) and (4.19) we get2 ZB(r=2)0w0(w h)dx Cr =2 ZB(r) [jr20wj2 + 1dx+ Cr2 2 ZB(r) jr0wj2dx : (4.20)Our next step is to estimate the Hessian term on the RHS of (4.20) by a terminvolving the Lapla ian, but on a larger ball. Su h an estimate follows from[GT, p. 234. More pre isely, for ea h > r, there is a universal onstant Csu h thatZB(r) jr20wj2dx C r4 " r4 ZB()(0w)2dx+ 2( r)4 ZB() jr0wj2dx# (4.21)Substituting (4.21) to (4.20), and ombining with (4.9) we get for all > r,ZB(r=2)(0w)2dx ZB(r=2)(0h)2dx+ Cr =2 r4 r4 ZB() (0w)2dx+ Cr =2 r4 2( r)4 ZB() jr0wj2dx+ Cr2 2 ZB(r) jr0wj2dx+ Cr =2 : (4.22)Sin e 0h is harmoni on B(r), (0h)2 is subharmoni there. The mean valueproperty for subharmoni fun tions then impliesZB(r=2)(0h)2dx ZB(r)(0h)2dx ZB(r)(0w)2dx : (4.23)Combining (4.22), (4.23), we getZB(r=2)(0w)2dx ZB(r)(0w)2dx+ C r =2 r4 r4 ZB()(0w)2dx+ C r =2 r4 2( r)4 ZB()(0w)2dx+ C r2 2 ZB(r)(r0w)2dx+ C r =2 : (4.24)35
We now multiply both sides of (4.24) by r2p; hoose p = 2 4 , and hoose = 2p=(4p)r, to get r2p ZB(r=2)(0w)2dx p ZB()(0w)2dx+ C r =2 ZB()(0w)2dx+ C r =4 ZB()(jr0wj2 + 1)dx (4.25)where C is a onstant depending on K and .We now turn our attention to estimating the gradient term in G(r). To this end,let v denote the harmoni extension of w to B(): 0v = 0 in B()v = w on B()then ZB(r=2) jr0wj2dx = ZB(r=2)r0w r0v dx+ ZB(r=2)r0w r0(w v)dx 12 ZB(r=2) jr0wj2dx+ 12 ZB(r=2) jr0vj2dx+ ZB(r=2)r0w r0(w v)dx ;)ZB(r=2) jr0wj2dx ZB(r=2) jr0vj2dx+ 2 ZB(r=2)r0w r0(w v)dx (4.27)As before, we estimate the ross-term in (4.27) via the S hwarz inequality:2 ZB(r=2)r0w r0(w v)dx 2ZB(r=2) jr0wj2dx1=2ZB(r=2) jr0(w v)j2dx1=2 2ZB() jr0wj2dx1=2ZB() jr0(w v)j2dx1=2 : (4.28)Sin e w v 2 W 2;20 (B()), by the Poin are inequality (P2) we haveZB() jr0(w v)j2dx . r2 ZB() [0(w v)2dx. r2 ZB()(0w)2dx : (4.29)36
Combining (4.28) and (4.29) we have2 ZB(r=2)r0w r0(w v)dx . rZB() jr0wj2dx1=2ZB()(0w)2dx1=2 r2 =8 ZB()(0w)2dx+ r =8 ZB() jr0wj2dx :(4.30)As for the rst term on the RHS of (4.27), we use the fa t that jr0vj2 is subhar-moni in B(), so by the mean value propertyZB(r=2) jr0vj2dx ZB()jr0vj2dx :Also, by the variational hara terization of the harmoni extensionZB() jr0vj2dx ZB() jr0wj2dxso that ZB(r=2) jr0vj2dx ZB() jr0wj2dx : (4.31)Combining (4.27), (4.30), and (4.31),ZB(r=2)jr0wj2dx (1 + Cr =8) ZB()jr0wj2dx+ Cr2 =8 ZB()(0w)2dx :Re all p = 2 4 , = 2p=(4p)r, thus r2 =8 Cr =8p, soZB(r=2)jr0wj2dx (1 + Cr =8) ZB() jr0wj2dx+ Cr =8p ZB()(0w)2dx : (4.32)Finally, ombining (4.25) and (4.32) we haveG(r=2) (1 + Cr =8)G() (4.33)The boundedness of G then follows from (4.33) by a standard iteration argu-ment. We will now apply a bootstrap argument to prove that for fun tions w satisfyingequation (1.2) and jrwj , jwj is a tually bounded. We start with a te hni allemma, the statement of whi h may be well known among experts, but we willin lude the proof here as we do not know an obvious referen e.37
Lemma 4.3. Suppose u is a C2 fun tion dened on a bounded domain in R4 ,and suppose that for some onstant 1 and 0 q < 2, RB(r)(0u)2dx 1r2+q forall balls B(r) of radius r in with r suÆ ient small. Then there exists a onstant 2 with RB(r) jr20uj2dx 2 2+q for all B(r) for r suÆ ient small.Proof. Fix x0 2 , B(R) = B(R; x0) and r < < R with R small so thatRB(R)(0u)2dx 1R2+q.Choose v so that 0v = 0 on B(R)v = u on B(R) :Let f = v u and apply inequality (4.21) to f ,ZB(r)jr20(v u)j2dx . r4 ZB() j0(v u)j2dx+ 2( r)4 ZB() jr0(v u)j2dx : (4.34)Observe that ZB()jr0(v u)j2dx ZB(R) jr0(v u)j2dx. R2 ZB(R) j0(v u)j2 1R2 R2+q :Thus from (4.34) we have for r < < R,ZB(r)jr20(v u)j2dx . r4R2+q + 1( r)4R6+q (4.35)To estimate RB(R) jr20vj2, we apply Theorem 2.1, and Remark 2.3 following Theorem2.1 in Giaquinta [G:ZB(r) jr20vj2dx . r4 ZB() jr20vj2dx. r4 ZB() jr20(v u)j2dx+ ZB() jr20uj2dx (4.36)Now apply (4.34) to v u on B(), hoose = r2 and sum up (4.35) and (4.36);then for r < R=2 we haveZB(r) jr20uj2dx . RR 2r4R2+q + rR4 ZB(R) jr20uj2dx+ rR4 R2+q :38
Dene (r) = RB(r) jr20uj2dx, then for ea h r R=4(r) A rR4 (R) +B R2+q (4.37)for some onstants A, B depending on 1. Applying Lemma 2.1 ([G, p. 86), wethen on lude that for ea h q < 2 there is a onstant = (A;B; q) so that(r) [A rR2+q (R) + Br2+q (4.38)for all r R=4. It follows from (4.38) that for R small and all r R=4(r) 2r2+qfor some onstant 2. Remark. It follows from (4.37) that if RB(r)(w)2 . r4 (i.e., jwj is bounded) thenjr20wj is bounded.We will apply an iterative argument to prove the following lemma.Lemma 4.4. Let w be as in Proposition 4.2; then there is some onstant so thatjr20!j .Proof. We remark rst that as a onsequen e of Proposition 4.2, jr0wj is bounded.We an now apply the argument in Proposition 2.1 to the biharmoni fun tion hwhi h agrees with w to rst order on B(r), and on lude that jh(x) h(x0)j . rfor all x 2 B(r) with x0 = rxjxj . Hen e jw hj . r on B(r).We now iterate the argument in (4.9), (4.10), (4.13), (4.15), (4.16) with theimproved estimates jw hj . r, jrwj and obtainZB(r=2)(0w)2 ZB(r=2)(0h)2+ ZB(r=2)(0w)21=2r ZB(r) [jr20wj2 + 11=2 (4.39)Applying Lemma 4.3 to the fun tion w with q = q1 = 2 p = =4; we haveR B(r)jr20wj2 r2+q1 for r suÆ iently small. Thus from (4.39), and (4.23) wehaveZB(r=2)(0w)2 ZB(r)(0w)2 + ZB(r=2)(0w)21=2 (r1+q1)1=2 : (4.40)>From a standard argument, we then on lude from (4.40) that for any " > 0, thereis a onstant C" > 0 with ZB(r)(0w)2 C" 1+q1" : (4.41)39
Choose " = q1=2, apply Lemma 4.3 with q = q2 = 1 + q1=2 > 1 and repeat thepro ess in (4.39) to getZB(r=2)(0w)2 ZB(r)(0w)2 + ZB(r=2)(0w)21=2 (r1+q2)1=2 ZB(r)(0w)2 + r q212 ZB(r) [(0w)2 + 1Hen e 0w is bounded, and by the remark following Lemma 4.3, jr20wj is bounded.Lemma 4.5. Let w be as in Proposition 4.2, then w 2 C1.Proof. The proof is immediate from Lemma 4.3 and the Euler equation (1.2) byellipti regularity.Proof of the Main Theorem. Let w be a minimizer of F . By Lemma 3.3, w is Holder ontinuous and satises (4.7). Regularity then follows from Lemma 4.4.
40
Referen es[B F. Bethuel; \On the singular set of stationary harmoni maps", Manus riptaMath., 78 (1993), pp 417-443.[BCY T. Branson, S-Y. A. Chang and P. C. Yang; \Estimates and extremal prob-lems for the log-determinant on 4-manifolds", Comm. Math. Physi s, Vol.149, No. 2, 1992, pp 241-262.[BO T. Branson and B. Orsted, Expli it fun tional determinants in four dimen-sions, Pro . A.M.S. 113 (1991), pp 669-682.[C Sun-Yung A. Chang; \ On Paneitz operator, a fourth order dierential op-erator in onformal geometry", preprint, 1997, to appear in the Pro eedingsConferen e for the 70th birthday of A.P. Calderon.[CQ1 Sun-Yung A. Chang and Jie Qing, "Zeta fun tional determinants on mani-folds with boundary ", Resear h announ ement, Math. Resear h Letters, 3(1996), pp 1-17.[CQ2 Sun-Yung A. Chang and Jie Qing, "The Zeta fun tional determinants onmanifolds with boundary II{Extremum metri s and ompa tness of isospe -tral set", to appear in JFA.[CY1 Sun-Yung A. Chang and Paul C. Yang, \Extremal Metri s of zeta Fun tionalDeterminants on 4-Manifolds, " Annals of Math. 142 (1995), pp 171-212.[CY2 Sun-Yung A. Chang and Paul C. Yang: "On uniqueness of solution of a n-thorder dierential equation in onformal geometry", Math. Resear h Letters,4 (1997), pp 91-102.[E C. L. Evans; \Partial regularity for stationary harmoni maps into spheres",Ar h. Rat. Me h. Anal. 116 (1991), pp 101-113[G M. Giaquinta, Multiple integrals in the al ulus of variation and nonlinearellipti systems, Annals of Math. Studies 105, Prin eton University Press1983.[GL N. Garofalo and F.{H. Lin, \Unique ontinuation for ellipti operators: Ageometri {variational approa h", Comm. Pure Appl. Math. 40 (1987), pp347{366.[GT D. Gilbarg and N. Trudinger, Ellipti PDE of se ond order, Springer Verlag,1983.[Gu M. Gursky, \ The Weyl fun tional, De Rham ohomology, and Kahler-Einstein metri s", preprint, 1996, to appear in the Annals of Math.[H1 F. Helein; \Regularity of weakly harmoni maps from a surfa e into a man-ifold with symmetries", Manus ripta Mathemati s, 70 (1991), pp 203-21841 Embed Size (px)
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