REGULARITY 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 de�ned 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 quanti�es the e�e 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 de�ned aslog det�g = log �1 + log �2 + � � � (0:1)where 0 < �1 � �2 � � � � are the non- zero eigenvalues of �g. While (0.1) is not wellde�ned a priori, there is a way of regularizing the de�nition 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 det�gwdet�g = � 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 + 2Kw�dA� �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) is�w + (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)satis�es (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 di�erentialoperators satisfying ertain \naturality" and \ onformal" assumptions whi h arerather te hni al in nature but are satis�ed, 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 de�ne the urvature invariantQ = 112���R� 3jRi j2 + R2�: (0:6)The onformal transformation rule for Q involves a fourth order onformallyinvariant operator �rst de�ned by Paneitz ([P℄): P = �2+ Æh 23Rg�2Ri 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 de�nitions, 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 jW�j2w dv � �Z jW�j2 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)2℄dv+ 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 in�nity then E(w) may not be integrable, hen e somethinglike assumption (A2) is ne essary.The Euler equation of F (see se tion 1) is�2w = 1(jr2wj2 � (�w)2 + Ri (rw;rw))+ 2�wjrwj2 + 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 de�ned 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; �w�n = 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 di�erential of u. If one imbeds Nk into someEu lidean spa e Rm , then the Euler equation of (0.14) is�u = 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 H�older 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 H�older 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 di�eren 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 pseudodi�erential 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 H�older 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 H�older 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 [B℄and [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 di�erent 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 H�older 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.

6

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)℄� 4�2�wjrwj2� 8�2r2w(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 satis�es�2w = 1(jr2wj2 � (�w)2 + Ri (rw;rw))+ 2�wjrwj2 + 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= ddt�Z (�w + t�')2dv+ Z (��w + �t�'+ �jrwj2 + 2�thr';rwi+ �t2jr'j2)2 dv+ Z (Dijriwrjw + 2tDijri'rjw + t2Dijri'rj'+E(w � �w + t('� �')) dv�����t=0= Z 2(1 + �2)�w�' dv+ Z 2��[2hr';rwi�w +�'jrwj2℄ dv+ Z [4�2hr';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';rwi�w +�'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';rwi�w 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;r�wi dvadding the above identity to (1.5) we get (1.4), and the lemma follows. �By (1.4), II = Z 2��[2hr';rwi�w +�'jrwj2℄dv= Z 4��'[jr2wj2 � (�w)2 +Ri (rw;rw)℄dv : (1.6)Finally, a simple integration by parts givesIII = Z '[�4�2�wjrwj2 � 8�2r2w(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 satis�es8>><>>:�20h = 0 in B(r)�h�n = � on �B(r)h = on �B(r) (2.1)where k�kLp(�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)jy2�B(r) = 0 (2.8)9

��nyG(x; y)jy2�B(r) = 0 (2.9)(�0)yG(x; y)jy2�B(r) = 4�r2 � jxj2r �2 1jx� yj4 (2.10)��ny (�0)yG(x; y)jy2�B(r) = 8�r2 � 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 satis�es (2.1), then for x 2 B(r),h(x) = I�B(r) ��ny (�0)yG(x; y) (y)ds(y)� I�B(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 , I�B(r) 1jx� yj2� ds(y) � C� 1�2��3x : (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 thatI�B(r) 1jx� yj2� ds(y) � C���2�x I�B(r) ds(y)� C���2�x r3� C���2�x (2�x)3� C� 1�2��3x :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) : 2k�1�x < jy � x0j � 2k�xg :Then by (2.15), y0 2 A0 ) jx� yj2� � C��2�x ;while learly IA0 ds(y) � �3x :Hen e IA0 1jx� yj2� ds(y) � C�� 12��3x : (2.16)It also follows from (2.15) thaty 2 Ak ) jx� yj2� � C�(2k�1�x)2� ;while IAk ds(y) � (2k�x)3 :Hen e IAk 1jx� yj2� ds(y) � C� 1(2k�1�x)2� IAk ds(y)� C�(2k�x)3(2k�1�x)�2�= C� 1�2��3x (23�2�)k : (2.17)Combining (2.16) and (2.17) we haveI�B(r) 1jx� yj2� ds(y) = 1Xk=0IAk 1jx� yj2� ds(y)� C� 1�2��3x "1 + 1Xk=1(23�2�)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 � I�B(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 = I�B(r) ��ny (�0)yG(x; y)( (y)� (x0))ds(y)II = � I�B(r)(�0)yG(x; y)�(y)ds(y) :By (2.1) and our assumption (2.3), we havejIj � �r2 � jxj2r �3 I�B(r) K2jy � x0j�jx� yj6 ds(y). K2�3x I�B(r) 1jx� yj6��where in the last line we have used the fa t that jx�yj � jx�x0j for all y 2 �B(r).Applying Lemma 2.3, we getjIj � K2�3x �C��3�(6��)x � . K2��x :Applying (2.10), we havejIIj � 4�r2 � jxj2r �2 I�B(r) j�(y)jjx� yj4 ds(y). �2xk�kLp(�B(r)) I�B(r) ds(y)jx� yj4p0! 1p 0where 1p + 1p0 = 1. Sin e p > 3, p0 > 34 , so by Lemma 2.3, we havejIIj . K1�2x ��3�4p0x � 1p0 = K1�1� 3px :Combining the estimates in I and II, we getjh(x)� h(x0)j . K2��x +K1�1� 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 satis�es (2.1). Then��h�0hj�B(r) = 2P3 + 2�T � 2�T� (2.18)�0hj�B(r) = 2�T � 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 satis�es (2.1) with8<: ' = �w�n = w : (2.21)ThenI�B(r) jr20hj2ds � C "I�B(r)(jr20wj2 + r�2jr0wj2)ds+ r�1 ZB(r)(�0h)2dx#(2.22)for some onstant C whi h does not depend on h or w.Proof. We havejr20hj2 . ��2h�n2�2 + jrT �h�n j2 + jrTrThj2 + r�2jr0hj2 :Sin e w = h and �h�n = �w�n on �B(r), this be omesjr20hj2 . ��2h�n2�2 + jrT �w�n j2+ jrTrThj2 + r�2jr0hj2� ��2h�n2�2 + jr20wj2 + jrTrThj2+ r�2jr0hj2 :The integrated Bo hner identity on �B(r) givesI�B(r) jrTrThj2ds . I�B(r)(�Th)2ds13

so that I�B(r)jr20hj2ds . I�B(r) ��2h�n2�2 ds+ I�B(r) �jr20wj2 + (�Th)2 + r�2jr0wj2� ds : (2.23)We now laimI�B(r) ��2h�n2�2 ds . I�B(r) jr20wj2ds+ I�B(r) r�2jr0wj2ds; (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 �w�n on �B(r) are given by 8>>><>>>: w =Xk akYk�w�n =Xk bkYk : (2.25)Using [CQ2, Lemma 3.3℄ we an write the expansion of h in B(r), and di�erentiationyields �2h�n2 j�B(r) =Xk f�k(k + 2)ak � (2k + 1)bkgYk :Therefore, I�B(r) ��2h�n2�2 ds �Xk fk2(k + 2)2a2k + (2k + 1)2b2kg : (2.26)Using (2.25) we also haveI�B(r)(�Tw)2ds =Xk k2(k + 2)2a2k ;I�B(r) �(��T + 1)1=2 �w�n �2 =Xk (k + 1)2b2k ;whi h when ompared to (2.26) givesI�B(r) ��2h�n2�2 ds . I�B(r)((�Tw)2 + �(��T + 1)1=2�w�n �2) ds. I�B(r) �jr20wj2 + r�2jr0wj2� ds :14

>From (2.23) and (2.24) we getI�B(r)jr20hj2ds � I�B(r) jr20wj2ds+ I�B(r)(�Th)2ds+ r�2 I�B(r) jr0wj2ds : (2.27)Writing �0j�B(r) in terms of �T we see that�0h = �2h�n2 + 3r �h�n +�Th= �2h�n2 + 3r �w�n +�Th)I�B(r)(�Th)2ds. I�B(r) "��2h�n2�2 + (�0h)2 + r�2��w�n�2# ds. I�B(r) �jr20wj2 + (�0h)2 + r�2jr0wj2� ds : (2.28)By (2.19),�0hj�B(r) = 2�Tw + 2f(��T + 1)1=2 + 1g�w�n)I�B(r)(�0h)2ds . I�B(r) �(�Tw)2 + jr20wj2 + r�2jr0wj2� ds. I�B(r) �jr20wj2 + r�2jr0wj2� ds :Substituting this into (2.28) givesI�B(r)(�Th)2ds. I�B(r) �jr20wj2 + r�2jr0wj2� ds : (2.29)Substituting (2.29) into (2.27) we getI�B(r)jr20hj2ds� I�B(r) �jr20wj2 + r�2jr0wj2� 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 + r�2jr0f j2)dx+ r I�B(r) �jr20f j2 + r�2jr0f 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, 12�0(jr0f j2) = jr20f j2 + hr0(�0f) ;r0fi : (2.32)Integrating by parts givesZB(r)hr0(�0f) ;r0fidx = ZB(r)�(�0f)2dx+ I�B(r)�0f �f�n : (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 = � I�B(r)�0f �f�n dsTo estimate I, we haveI = 12 I�B(r) ��n(jr0f j2)ds. I�B(r) jr20f jjr0f jds. r I�B(r) �jr20f j2 + r�2jr0f j2� ds (2.35)We an estimate II asjIIj . I�B(r)(�0f)2 ds!1=2 I�B(r) ��f�n�2 ds!1=2. I�B(r) jr20f j2 ds!1=2 I�B(r) jr0f j2 ds!1=2. r I�B(r) jr20f j2ds+ r�1 I�B(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 I�B(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)) thenI�B(r)(f � �f)2 ds . r2 I�B(r) jrT0 f j2 dswhere �f = 1j�B(r)j H�B(r) f ds(P4) If f 2W 2;2(�B(r)) thenI�B(r) jrT0 f j2 ds . r2 I�Br (�T0 f)2 ds :17

x3. H�older Continuity of MinimizersIn this se tion we derive a preliminary regularity result for minimizers of (1.1);namely, we show that they must be H�older 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 � � �dx4�0 = 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(�i�jw + �kij�kw) (3.1)= (1 + O(r2))(�0w) +O(r2)jr20wj+ O(r)jr0wjjr2wj2 = (1 + O(r2))jr20wj2 + O(r2)jr0wj2To prove H�older 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 satis�es 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 H�older's inequality.To this end, de�neD(r) = ZB(r)[jr20wj2 + jr0wj4 + w2℄dx; (3.4)E(r) = D(r) + T (r);where T (r) = ZB(r) jxj�2jr0wj2dx: (3.5)Let us begin with some preliminary remarks whi h explain when T (r) makes sense.First de�ne T�(r) = ZB(r)�B(�) jxj�2jr0wj2dx:Then T�(r) is well de�ned for � > 0. Writing T�(r) in polar oordinates we haveT�(r) = Z r� t�2 I�B(t) jr0wj2ds! dt:Integrating by parts we getT�(r) = �r�1 I�B(r) jr0wj2ds+ ��1 I�B(�) jr0wj2ds+ Z r� t�1 ddt I�B(t) jr0wj2ds! dt : (3.6)Note that ddt I�B(t) jr0wj2ds! = ddt I�B(1) jr0w(t; �)j2t3d�where d� is the surfa e measure on �B(1). Thusddt I�B(t) jr0wj2ds! = 3t�1 I�B(1) jr0w(t; �)j2t3d�+ I�B(1) � ddt jr0w(t; �)j2� t3d�:19

Substituting this ba k into (3.6) we getT�(r) = �r�1 I�B(r) jr0wj2ds+ ��1 I�B(�) jr0wj2ds+ 3T�(r) + Z r� t�1 I�B(1) � ddt jr0w(t; �)j2� t3d�dt:Hen e,2T�(r) � r�1 I�B(r) jr0wj2ds+ 2 ZB(r)�B(�) jxj�1jr20wjjr0wjdx� r�1 I�B(r) jr0wj2ds+ 2 ZB(r)�B(�) jxj�2jr0wj2dx!1=2 ZB(r) jr20wj2dx!1=2� r�1 I�B(r) jr0wj2ds+ T�(r) + ZB(r) jr20wj2dx:Therefore, T�(r) � r�1 I�B(r) jr0wj2ds+ ZB(r) jr20wj2dx:Taking the limit as �! 0 we have provedLemma 3.1. If H�B(r) jr0wj2ds <1, then T (r) <1 andT (r) � r�1 I�B(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 di�erentiable 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 satis�es (3.9). For, given � 2 (0; 1), there is a measurableset �� � (r0=2; r0) su h that rD0(r) � 2��1D(r0) (3.12)for all r 2 ��, and j��j = (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 inf�2[r;2r℄�D0(�) � 32 rD0(r) : (3.13)Note that by taking � = 34 we may on lude from (3.12) thatinf�2[r;2r℄�D0(�) � 83 D(2r) :Hen e r1D0(r1) � 32 �83D(2r)� = 4D(2r) � 4 �C :So by Proposition 3.2, E(r1) � Ar1E0(r1) +Ar�1 : (3.14)We now observe thatr1E0(r1) = r1D0(r1) + r1T 0(r1)= r1D0(r1) + r1�1 I�B(r1) jr0wj2ds. r1D0(r1) + r1 I�B(r1) jr0wj4ds!1=2. r1D0(r1) + (r1D0(r1))1=2. 4 �C + �4 �C�1=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) + Cr�1 (By (3.14))� 32CrE0(r) + Cr�1 (By (3.13))� 32CrE0(r) + C(2r)�� 2CrE0(r) + 4Cr�for some onstant C = C( �C;A). �Sin e (3.11) holds for all r suÆ iently small, we an integrate the di�erentialinequality 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)�h�n = �w�n 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 de�ned.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 + jr0wj4℄dx . ZB(r)[(�0h)2 + jr0hj4℄dx+ r� : (3.18)22

Proof. Sin e w is extremal, F [w℄ � F [h℄ : (3.19)Given �M , let us de�neF 1(') = Z[(�')2 + (��'+ �jr'j2)2℄dv ;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 2��xy � �(�2 + 12)x2 � 2�2�22�2+1y2 we haveF 1B(w) = ZB[(1 + �2)(�w)2 + 2���wjrwj2 + �2jrwj4℄dv� ZB �12(�w)2 + �21 + 2�2 jrwj4� dv& ZB[(�w)2 + jrwj4℄dv (3.20)Likewise, F 1B(h) � ZB [(1 + 2�2)(j�h)j2 + 2�2jrhj4℄dv. ZB [(�h)j2 + jrhj4℄dv : (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� �wjdv�1=2 + �ZB e2a2jh��hjdv�1=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 e2a2jh��hjdv � C ;hen e jZ(w; h)j . r� (3.22)with � = 2.Combining (3.20) - (3.23) we getZB [(�w)2 + jrwj4℄dv . ZB [j�hj2 + jrhj4℄dV + 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 h�20hdx = ZB(�0h)2dx+ I�B h ��n(�0h)ds� I�B �0h�h�nds (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 = I�B �h ��n (�0h) ds ; (3.26)I2 = I�B �0h �h�n ds : (3.27)Let �T and rT denote the Eu lidean tangential Lapla ian and gradient on �B.By Proposition 2.4 ��n�0h = 2P3w � 2�Tw + 2�T �w�n (3.28)where P3 is the pseudo-di�erential 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 justi�es the inte-gration in (3.26).In any ase we have�I1 = I�B 2w�P3w ��Tw +�T �w�n� ds= Ia1 + Ib1 + I 1 :Using (3.29) and integrating by parts,Ia1 = I�B 2wP3w ds= 2 I�B �(��T + 1)1=2w�Tw ds= �2 I�B[(��T + 1)1=2w � �1℄�Tw ds ;where �1 = 1j�Bj I�B(��T + 1)1=2w ds ;j�Bj = I�B ds :By the S hwarz inequalityjIa1 j � 2�I�B h(��T + 1)1=2w � �1i2 ds�1=2�I�B(�Tw)2ds�1=225

and by the Poin air�e inequality (P3)I�B[(��T + 1)1=2w � �1℄2ds . r2 I�B jrT (��T + 1)1=2wj2ds ;so that jIa1 j . r�I�B jrT (��T + 1)1=2wj2ds�1=2�I�B(�Tw)2ds�1=2. r I jrTrTwj2ds. r I�B jr20wj2ds+ r�1 I�B jr0wj2ds : (3.30)Using a similar argument, Ib1 = �2 I�B w�Tw ds= �2 I�B(w � �2)�Tw dswhere �2 = 1j�Bj I�B w ds :Then using the Poin air�e inequality (P4) we havejIb1j � 2�I�B(w � �2)2ds�1=2�I (�Tw)2ds�1=2. �r4 I�B(�Tw)2ds�1=2�I�B(�Tw)2ds�1=2. r2 I�B jr20wj2ds+ I�B jr0wj2ds : (3.31)Finally, I 1 = 2 I�B w�T ��w�n� ds= 2 I�B �Tw�w�n dsHen e, jI 1j � 2�I�B(�Tw)2 ds�1=2�I�B jr0wj2ds�1=2. �I�B jr20wj2 ds+ r�2 I�B jr0wj2ds�1=2 �I�B jr0wj2ds�1=2. r I�B jr20wj2 ds+ r�1 I�B jr0wj2ds (3.32)26

Combining (3.30) - (3.32) we on ludejI1j . r I�B jr20wj2 ds+ r�1 I�B jr0wj2ds : (3.33)To estimate I2, we appeal on e more to Proposition 2.4 and write�0h = 2�Tw � 2 h(��T + 1)1=2 + 1i �w�nso that I2 = I�B �0h�h�n ds= I�Bf2�Tw � 2[(��T + 1)1=2 + 1℄�w�n g �w�n ds= Ia2 + Ib2 :Noti e jIa2 j = jI 1j . r I�B jr20wj2 ds+ r�1 I�B jr0wj2ds : (3.34)Also, Ib2 = I�B �2 h(��T + 1)1=2 + 1i �w�n �w�n dsHen e, jIb2j . I�B �jr20wj+ r�1jr0wj� jr0wjds. r I�B jr20wj2 ds+ r�1 I�B jr0wj2 ds: (3.35)Therefore, jI2j � jIa2 j+ jIb2j. r I�B jr20wj2 ds+ r�1 I�B jr0wj2 ds: (3.36)Combining (3.33) and (3.36) we getI . r I�B jr20wj2 ds+ r�1 I�B 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 get�ZB jr0hj4 dx�1=2 . ZB jr20hj2 dx+ r�1 I�B 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 + ��h�n�2= jr0wj2:Therefore, �ZB jr0hj4dx�1=2 . ZB jr20hj2dx+ r�1 I�B jr0wj2 ds: (3.40)By the assumption (3.9) rD0(r) � 4 �CTherefore, r I�B jr0wj4 � 4 �C:Hen e, r�1 I�B jr0wj2ds � r�3=2�r I�B jr0wj4ds�1=2�I�B ds�1=2 � C:It then follows from (3.8) and (3.17) that�ZB jr0hj4dx�1=2 . 1: (3.41)>From (3.40) and (3.41) we on ludeZB jr0hj4dx . ZB jr20hj2dx+ r�1 I�B 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+ r�1 I�B jr0wj2 ds+ r I�B 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 I�B jr20wj2 ds+ r�1 I�B jr0wj2 ds :Finally, by (3.37) we on ludeII = ZB jr0hj4 dx . r I�B jr20wj2 ds+ r�1 I�B jr0wj2 ds : (3.44)Combining (3.37), (3.44), and (3.18) we haveZB[(�0w)2 + jr0wj4℄ dx . r I�B jr20wj2 + r�1 I�B jr0wj2 ds+ r�: (3.45)Re all that D(r) = ZB [jr20wj2 + jr0wj4 + w2℄ dx :Using the Sobolev inequality (S2),ZB w2dx � jBj1=2�ZB w4dx�1=2. r2�ZB jr0wj2dx+ r�1 I�B w2 ds�. r2 ZB jr0wj4dx+ r I�B w2ds+ r6. ZB jr0wj4dx+ r I�B w2 ds+ r� :Therefore, D(r) . ZB [jr20wj2 + jr0wj4℄dx+ r I�B w2ds+ r� : (3.46)Now, E(r) = D(r) + T (r);so by Lemma 3.1,E(r) . ZB[jr20wj2 + jr0wj4℄dx+ r I�B [r�2jr0wj2 + w2℄ds+ r�:Taking f = w in Proposition 2.6,E(r) . I�B[j�0wj2 + jr0wj4℄dx+ r I�B[jr20wj2 + r�2jr0wj2 + w2℄ds+ r�:And therefore by (3.45)E(r) . r I�B[jr20wj2 + r�2jr0wj2 + w2℄ds. rE0(r) + r� ;whi h ompletes the proof of Proposition 3.2. �29

x4. Classi al regularity from H�older 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 H�older 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 have�����ZB(r)�0w�0'dx����� � C(r + supB(r) j'j) ZB(r)[jr20wj2 + jr0wj4 + 1℄dx+ Cr(1 + supB(r) j'j) ZB(r)[1 + jr20'j2℄dx : (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)j℄dv :By assumption (A3), jE0(w � �w)j � a1exp a2jw � �wj. Sin e w 2 C we on lude����Z �w�' dv���� . Z j'j[jr2wj2 + jrwj4 + 1℄dv : (4.2)Using (3.1) and the fa t that supp ' � B(r) we haveZ j'j[jr2wj2 + jrwj4 + 1℄dv= 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 + 1℄dx (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 �0w�0'dx = Z �w �' dv+ O(r) Z [jr20wjjr20'j+ jr20wjjr0'j+ jr0wjjr20'j℄dx) ����Z �0w�0' dx���� � ����Z �w �' dv����+ Cr Zsupp'[jr20wj2 + jr20'j2 + jr0wj2 + jr0'j2℄dx :Sin e supp ' � B(r), Z jr0'j2dx = Z �'�0' dx� supB(r) j'j ZB(r) j�0'j dx� C supB(r) j'j ZB(r)[1 + jr20'j2℄dx :Therefore, ����Z �0w�0' dx���� � ����Z �w�' dV ����+ Cr ZB(r)[jr20wj2 + jr0wj4 + 1℄dx+ Cr(1 + supB(r) j'j) Z [1 + jr20'j2℄dx : (4.4)Combining (4.2), (4.3), and (4.4) we get (4.1). �>From now on let us adopt the onvention that R�B(s)dx = (vol B(s))�1 RB(s) dx.De�ne G(r) = Z�B(r) [rp(�0w)2 + jr0wj2 + 1℄dx ; (4.5)where 0 < p < 2 will be spe i�ed 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 + w2℄dx � 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 satis�esjr0wj � 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 de�ned 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 satis�es ' 2W 2;2 \ C 0 (B(r)).Our estimate of G(r) begins with an estimate ofZ�B(r=2)(�0w)2dx = Z�B(r=2)�0w(�0w ��0h) dx+ Z�B(r=2)�0w �0h dx� Z�B(r=2)�0w �0(w � h) dx+ �Z�B(r=2)(�0w)2dx�1=2�Z�B(r=2)(�0h)2dx�1=2� Z�B(r=2)�0w�0(w � h)dx+ 12 Z�B(r=2)(�0w)2dx+ 12 Z�B(r=2)(�0h)2dx) Z�B(r=2)(�0w)2dx � Z�B(r=2)(�0h)2dx+ 2 Z�B(r=2)�0w �0(w � h)dx : (4.9)Examining the se ond term on the RHS of (4.9),Z�B(r=2)�0w�0(w � h)dx� �Z�B(r=2)(�0w)2dx�1=2�Z�B(r=2) [�0(w � h)℄2dx�1=2� C �Z�B(r)(�0w)2dx�1=2�Z�B(r) [�0(w � h)℄2dx�1=2 (4.10)32

Noti e that Z�B(r) [�0(w � h)℄2dx= Z�B(r) [(�0w)2 � 2�0w�0h+ (�0h)2℄dx= Z�B(r) [2�0w �0'+ (�0h)2 � (�0w)2℄dx : (4.11)Re all the variational hara terization of h as in (3.16), whi h in parti ular impliesZ�B(r)(�0h)2dx � Z�B(r)(�0w)2dx : (4.12)Therefore by (4.11), (4.1) we haveZ�B(r) [�0(w � h)℄2dx � 2 Z�B(r)�0w �0' dx� C r + supB(r) j'j!Z�B(r) [jr20wj2 + jr0wj4 + 1℄dx+ Cr 1 + supB(r) j'j!Z�B(r) [1 + jr20'j2℄dx : (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):Z�B(r) jr20hj2dx � Z�B(r) jr20wj2dx :Therefore, Z�B(r) jr20'j2dx = Z�B(r) jr20(w � h)j2dx� 2 Z�B(r)(jr20wj2 + jr20hj2)dx� 4 Z�B(r) jr20wj2dx : (4.15)33

Combining (4.13) - (4.15) we on ludeZ�B(r) [�0(w � h)℄2dx � Cr Z�B(r) [jr20wj2 + jr0wj4 + 1℄dx : (4.16)Therefore, by (4.9) and (4.16),2 Z�B(r=2)�0w�0(w � h)dx� Cr =2�Z�B(r)(�0w)2dx�1=2�Z�B(r) [r20wj2 + jr0wj4 + 1℄dx�1=2� Cr =2 Z�B(r) [jr20wj2 + jr0wj4 + 1℄dx : (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:�Z�B(r) jr0wj4dx�1=2 . r2 Z�B(r) jr0jr0wjj2dx+ Z�B(r) jr0wj2dx. r2 Z�B(r) jr20wj2dx+ Z�B(r) jr0wj2dx : (4.18)By (4.6) the RHS of (4.18) an be estimated as follows:r2 Z�B(r) jr20wj2dx+ Z�B(r) jr0wj2dx. r�2 "ZB(r) jr20wj2dx+ r�2 ZB(r) jr0wj2dx#. r�2 "Kr4 + r�2 �Z jr0wj4dx�1=2 jB(r)j1=2#. r�2[Kr4 + CK1=2r2 ℄. r2 �2 :Therefore Z�B(r) jr0wj4dx . �r2 Z�B(r) jr20wj2dx+ Z�B(r) jr0wj2dx�2� r2 �2 �r2 Z�B(r) jr20wj2dx+ Z�B(r) jr0wj2dx�� r2 Z�B(r) jr20wj2dx+ r2 �2 Z�B(r) jr0wj2dx : (4.19)34

Combining (4.17) and (4.19) we get2 Z�B(r=2)�0w�0(w � h)dx � Cr =2 Z�B(r) [jr20wj2 + 1℄dx+ Cr2 �2 Z�B(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 thatZ�B(r) jr20wj2dx� C ��r�4 "� ��� r�4 Z�B(�)(�0w)2dx+ �2(�� r)4 Z�B(�) jr0wj2dx# (4.21)Substituting (4.21) to (4.20), and ombining with (4.9) we get for all � > r,Z�B(r=2)(�0w)2dx � Z�B(r=2)(�0h)2dx+ Cr =2 ��r�4 � ��� r�4 Z�B(�) (�0w)2dx+ Cr =2 ��r�4 �2(�� r)4 Z�B(�) jr0wj2dx+ Cr2 �2 Z�B(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 impliesZ�B(r=2)(�0h)2dx � Z�B(r)(�0h)2dx� Z�B(r)(�0w)2dx : (4.23)Combining (4.22), (4.23), we getZ�B(r=2)(�0w)2dx � Z�B(r)(�0w)2dx+ C r =2 ��r�4� ��� r�4 Z�B(�)(�0w)2dx+ C r =2 ��r�4 �2(�� r)4 Z�B(�)(�0w)2dx+ C r2 �2 Z�B(r)(r0w)2dx+ C r =2 : (4.24)35

We now multiply both sides of (4.24) by � r2�p; hoose p = 2 � 4 , and hoose� = 2p=(4�p)r, to get� r2�p Z�B(r=2)(�0w)2dx � �p Z�B(�)(�0w)2dx+ C r =2 Z�B(�)(�0w)2dx+ C r =4 Z�B(�)(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 Z�B(r=2) jr0wj2dx = Z�B(r=2)r0w � r0v dx+ Z�B(r=2)r0w � r0(w � v)dx� 12 Z�B(r=2) jr0wj2dx+ 12 Z�B(r=2) jr0vj2dx+ Z�B(r=2)r0w � r0(w � v)dx ;)Z�B(r=2) jr0wj2dx � Z�B(r=2) jr0vj2dx+ 2 Z�B(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 Z�B(r=2)r0w � r0(w � v)dx� 2�Z�B(r=2) jr0wj2dx�1=2�Z�B(r=2) jr0(w � v)j2dx�1=2� 2�Z�B(�) jr0wj2dx�1=2�Z�B(�) jr0(w � v)j2dx�1=2 : (4.28)Sin e w � v 2 W 2;20 (B(�)), by the Poin are inequality (P2) we haveZ�B(�) jr0(w � v)j2dx . r2 Z�B(�) [�0(w � v)℄2dx. r2 Z�B(�)(�0w)2dx : (4.29)36

Combining (4.28) and (4.29) we have2 Z�B(r=2)r0w � r0(w � v)dx . r�Z�B(�) jr0wj2dx�1=2�Z�B(�)(�0w)2dx�1=2� r2� =8 Z�B(�)(�0w)2dx+ r =8 Z�B(�) 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 propertyZ�B(r=2) jr0vj2dx � Z�B(�)jr0vj2dx :Also, by the variational hara terization of the harmoni extensionZ�B(�) jr0vj2dx � Z�B(�) jr0wj2dxso that Z�B(r=2) jr0vj2dx � Z�B(�) jr0wj2dx : (4.31)Combining (4.27), (4.30), and (4.31),Z�B(r=2)jr0wj2dx � (1 + Cr =8) Z�B(�)jr0wj2dx+ Cr2� =8 Z�B(�)(�0w)2dx :Re all p = 2� 4 , � = 2p=(4�p)r, thus r2� =8 � Cr =8�p, soZ�B(r=2)jr0wj2dx � (1 + Cr =8) Z�B(�) jr0wj2dx+ Cr =8�p Z�B(�)(�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 � , j�wj 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 de�ned 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 . � ��� r�4 ZB(�) j�0(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) j�0(v � u)j2 � 1R2 �R2+q :Thus from (4.34) we have for r < � < R,ZB(r)jr20(v � u)j2dx . � ��� r�4R2+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 . � r��4 ZB(�) jr20vj2dx. � r��4 �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� 2r�4R2+q + � rR�4 ZB(R) jr20uj2dx+ � rR�4 R2+q :38

De�ne �(r) = RB(r) jr20uj2dx, then for ea h r � R=4�(r) � A� rR�4 �(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� rR�2+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., j�wj 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 obtainZ�B(r=2)(�0w)2 � Z�B(r=2)(�0h)2+ �Z�B(r=2)(�0w)2�1=2�r Z�B(r) [jr20wj2 + 1℄�1=2 (4.39)Applying Lemma 4.3 to the fun tion w with q = q1 = 2 � p = =4; we haveR� B(r)jr20wj2 � r�2+q1 for r suÆ iently small. Thus from (4.39), and (4.23) wehaveZ�B(r=2)(�0w)2 � Z�B(r)(�0w)2 + �Z�B(r=2)(�0w)2�1=2 (r�1+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 Z�B(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 getZ�B(r=2)(�0w)2 � Z�B(r)(�0w)2 + �Z�B(r=2)(�0w)2�1=2 (r�1+q2)1=2� Z�B(r)(�0w)2 + r q2�12 Z�B(r) [(�0w)2 + 1℄Hen 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 H�older ontinuous and satis�es (4.7). Regularity then follows from Lemma 4.4.�

40

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