Hessian Estimates for the Sigma-2 Equation in Dimension 3

Transcription

1 Hessian Estimates for the Sigma- Euation in imension MICAH WARREN University of Washington AN YU YUAN University of Washington Abstract We derive a priori interior Hessian estimates for the special Lagrangian euation in dimension. c 008 Wiley Periodicals, Inc. Introduction In this article, we derive an interior a priori Hessian estimate for the euation (.). u/ C C in dimension, where i are the eigenvalues of the Hessian u: We attack (.) via its special Lagrangian euation form (.) arctan i i with n and : Euation (.) stems from the special Lagrangian geometry [4]. The Lagrangian graph.x; u.x// R n R n is called special when the phase or the argument of the complex number. C p /. C p n / is constant ; and it is special if and only if.x; u.x// is a (volume minimizing) minimal surface in R n R n [4, theorem., prop..7]. We state our result in the following: THEOREM. Let u be a smooth solution to (.) on B R.0/ R : Then we have j juj u.0/j C./ exp C./ max : B R.0/ By Trudinger s gradient estimates for k -euations [9], we can bound u in terms of the solution u in B R.0/ as j juj u.0/j C./ exp C./ max B R.0/ R 6 : Communications on Pure and Applied Mathematics, Vol. LXII, (009) c 008 Wiley Periodicals, Inc. R

2 06 M. WARREN AN Y. YUAN One immediate conseuence of the above estimates is a Liouville-type result for global solutions with uadratic growth to (.); namely, any such solution must be uadratic (cf. [5, 6]). Another conseuence is the regularity (analyticity) of the C 0 viscosity solutions to (.) or (.) with n and. In the 950s, Heinz [5] derived a Hessian bound for. u/ det. u/, the two-dimensional Monge-Ampère euation, which is euivalent to (.) with n and. In the 970s Pogorelov [8] constructed his famous counterexamples, namely, irregular solutions to three-dimensional Monge- Ampère euations. u/ det. u/ I see generalizations of the counterexamples for k -euations with k in [0]. Hessian estimates for solutions with certain strict convexity constraints to Monge-Ampère euations and k -euations (k ) were derived by Pogorelov [8] and Chou and Wang [], respectively, using the Pogorelov techniue. Urbas [, ] and Bao and Chen [] obtained (pointwise) Hessian estimates in terms of certain integrals of the Hessian, respectively, for k -euations and the special Lagrangian euation (.) with n and, The heuristic idea of the proof of Theorem. is as follows: The function b ln p C max is subharmonic so that b at any point is bounded by its integral over a ball around the point on the minimal surface by Michael and Simon s mean value ineuality [6]. This special choice of b is not only subharmonic but, even stronger, satisfies a Jacobi ineuality. This Jacobi ineuality leads to a bound on the integral of b by the volume of the ball on the minimal surface. Taking advantage of the divergence form of the volume element of the minimal Lagrangian graph, we bound the volume in terms of the height of the special Lagrangian graph, which is the gradient of the solution to euation (.). Now the challenging regularity problem for sigma- euations in dimension 4 and higher still remains open to us. Notation. i j i u, u ij u, etc., but ;:::; n and b ln C ; b ln C C ln C ; do not represent the partial derivatives. Further, h ij k will denote (the second fundamental form) h ij k C i C j C k u ij k ; when u is diagonalized. Finally, C.n/ will denote various constants depending only on dimension n:

3 HESSIAN ESTIMATES 07 Preliminary Ineualities Taking the gradient of both sides of the special Lagrangian euation (.), we have (.) i;j g ij.x; u.x// 0; where.g ij / is the inverse of the induced metric g.g ij / I C u u on the surface.x; u.x// R n R n : Simple geometric manipulation of (.) yields the usual form of the minimal surface euation 4 g.x; u.x// 0; where the Laplace-Beltrami operator of the metric g is given by 4 g p det g i p det gg j : Because we are using harmonic coordinates 4 g x 0; we see that 4 g also euals the linearized operator of the special Lagrangian euation (.) at u; 4 g i;j g ij : The gradient and inner product with respect to the metric g are hr g v; r g wi g r g v g k v k ; ; k i;j g nk v k ; k We begin with some geometric calculations. g ij v i w j ; in particular, jr g vj hr g v; r g vi g : LEMMA. Let u be a smooth solution to (.). Suppose that the Hessian u is diagonalized and theeigenvalue is distinct from all other eigenvalues of u at point p: Set b ln C near p: Then we have at p (.) jr g b j k h k

4 08 M. WARREN AN Y. YUAN and (.) (.4) (.5) 4 g b. C /h C X k> C X k> C X k>j> C k k k C k C. C k / k h k h kk C k C C j C. j C k / h kj k : j PROOF: We first compute the derivatives of the smooth function b near p: We may implicitly differentiate the characteristic euation det. u I/ 0 near any point where is distinct from the other eigenvalues. Then we get at e u ee u C X.@ eu k / ; k k> with arbitrary unit vector e R n : Thus we have (.) at p jr g b j g kk k u h k ; k k where we used the notation h ij k p g iip g p jj g kk u ij k : ee ee ln C ee C. C.@ e / ; / we conclude that at ee b ee u C X.@ eu k / C k. C.@ eu / k> / and (.6) 4 g b b g u C X.u k/ C k k>. C / g u :

5 HESSIAN ESTIMATES 09 Next we substitute the fourth-order derivative u in the above by lower-order derivative terms. ifferentiating the minimal surface euation (.) P n ;ˇ g ˇ u j ˇ 0; we obtain (.7) 4 g u ij where we used ;ˇ g ˇ u ji i g i ı ı C ;ˇ ;ˇ;;ı i g ˇ u j ˇ i g ı g ıˇ u j ˇ g gˇˇ. C ˇ /u ˇ i u ˇj ; u " u "ı u ıi. C ı / " with diagonalized u: Plugging (.7) with i j into (.6), we have at p 4 g b C g gˇˇ. C ˇ /u ˇ C X u k g k C ;ˇ. C / g u C ˇ /h ;ˇ. ˇ C X k> C. /h ; k>. C k / k h k where we used the notation h ij k p g iip g p jj g kk u ij k : Regrouping those terms h ~~ ;h ~ ; and h ~ in the last expression, we have 4 g b. /h C h C X. C k / h kk k k> C X. /h k C X. k C /h k C X. C k / h k k k> k> k> C X k>j>. j C k /h jk C X j;k>; j k. C k / k h jk :

6 0 M. WARREN AN Y. YUAN After simplifying the above expression, we have the second formula in Lemma.. LEMMA. Let u be a smooth solution to (.) with n and : Suppose that the ordered eigenvalues of the Hessian u satisfy > at point p: Set b ln C max ln C : Then we have at p (.8) 4 g b jr gb j : PROOF : We assume that the Hessian u is diagonalized at point p: Step. Recall i arctan i. ; / and C C : It is easy to see that >0and i C j 0 for any pair. Conseuently, >0and i C j 0 for any pair of distinct eigenvalues. It follows that (.5) in the formula for 4 g b is positive; then from (.) and (.4) we have the ineuality (.9) 4 g b h C X k X h kk C C k h k k : k k> k> Combining (.9) and (.) gives (.0) 4 g b jr gb j h C X k> k h kk k X C k>. C k /. k / h k : Step. We show that the second term on the right-hand side of (.0) is nonnegative. Note that C k C : We only need to show that C 0 in the case that <0or euivalently <0:From C C ; we have > C C C : It follows that then tan C >tan I (.) C >0:

7 HESSIAN ESTIMATES Step. We show that the first term in (.0) is nonnegative by proving (.) h C h C h 0: We only need to show it for < 0: irectly from the minimal surface euation (.) h C h C h 0I we bound h.h C h / h C h C : It follows that h C h C h h C h C The last term becomes C C >0:. / The above ineuality is from the observation Y p Re C i 0 i C : for > C C : Therefore (.) holds. We have proved the pointwise Jacobi ineuality (.8) in Lemma.. LEMMA. Let u be a smooth solution to (.) with n and : Suppose that the ordered eigenvalues of the Hessian u satisfy > at point p: Set b ln C C ln C : Then b satisfies at p (.) 4 g b 0: Further, suppose that in a neighborhood of p: Then b satisfies at p (.4) 4 g b jr gb j :

8 M. WARREN AN Y. YUAN PROOF: We assume that Hessian u is diagonalized at point p: We may use Lemma. to obtain expressions for both 4 g ln C and 4 g ln C ; whenever the eigenvalues of u are distinct. From (.), (.4), and (.5), we have (.5) 4 g ln C C4 g ln C. C /h C X. C k / h kk k k> C X C C C k h k k k> C C C C C. C / h C. C /h C X. C k / h kk k k C X C C C k h k k k C C C C C. C / h : The function b is symmetric in and ; thus b is smooth even when provided that >. We simplify (.5) to the following, which holds by continuity wherever > : (.6) (.7) (.8) (.9) 4 g b. C /h C. C C /h C C C. C C /h C. C /h C h C C C. C / h C C. C / C C C C C. C / C. C / h : h h Using the assumption coupled with the relations >0, i C j >0, and derived in the proof of Lemma., we see that (.9) and (.8) are nonnegative. We only need to justify the nonnegativity of (.6) and

9 HESSIAN ESTIMATES (.7) for <0:From the minimal surface euation (.), we know h.h C h / C h C h C C : It follows that (.7). C /h C h C h. C /h C h C C C The last term becomes C C C : 0:. C / Thus (.7) is nonnegative. Similarly, (.6) is nonnegative. We have proved (.). Next we prove (.4), still assuming u is diagonalized at point p: Plugging into (.6), (.7), and (.8), we get 4 g b h C h C h C h C h C h C C.h C h /: ifferentiating the eigenvector euations in the neighborhood where. u/u C U;. u/v C V; and. u/w W; we see that u e u e for any e R at point p: Using the minimal surface euation (.), we then have h k h k h k at point p: Thus 4 g b C h C C h C C h :

10 4 M. WARREN AN Y. YUAN The gradient jr g b j has the expression at p X jr g b j g kk k u C k u k X h k : k Thus at p 4 g b jr gb j C h C C h C C h 0; where we again used C > 0 from (.). We have proved (.4) of Lemma.. PROPOSITION.4 Let u be a smooth solution to the special Lagrangian euation (.) with n and on.0/ R : Set n o b max ln C max ;K with K C ln C tan. 6 /: Then b satisfies the integral Jacobi ineuality (.0) hr g ';r g bi g dv g 'jr g bj dv g for all nonnegative ' C 0./: PROOF: Ifb ln p C max is smooth everywhere, then the pointwise Jacobi ineuality (.8) in Lemma. already implies the integral Jacobi (.0). It is known that max is always a Lipschitz function of the entries of the Hessian u: Now u is smooth in x; so b ln p C max is Lipschitz in terms of x: If b (or euivalently max / is not smooth, then the two largest eigenvalues.x/ and.x/ coincide, and b.x/ b.x/; where b.x/ is the average b ln C C ln C : We prove the integral Jacobi ineuality (.0) for a possibly singular b.x/ in two cases. Set S fx j.x/.x/g:

11 HESSIAN ESTIMATES 5 Case. S has measure zero. For small >0;let nfx j b.x/ Kg nfx j b.x/ Kg;./ fx j b.x/ b.x/ > b.x/ C g\;./ fx j b.x/ b.x/ b.x/ < b.x/ C g\: Now b.x/ b.x/ is smooth in./: We claim that b.x/ is smooth in./: We know b.x/ is smooth wherever.x/ >.x/: If (the Lipschitz) b.x/ is not smooth at x./; then ln C ln C ln C ln C tan. 6 / C ; by the choice of K: For small enough, wehave > tan. 6 / and a contradiction Note that hr g ';r g bi g dv g. C C /.x /> : hr g ';r g bi g dv g lim!0 C./ hr g ';r g bi g dv g C./ hr g ';r g.b C /i g dv g : By the smoothness of b in./ and b in./; and also ineualities (.8) and (.), we have hr g ';r g bi g dv g C hr g ';r g.b C /i g dv g./ '@ g bda g C./ ' 4 g b dv C./ '@ g.b C /da g C ' 4 g.b C /dv

12 6 M. WARREN AN Y. YUAN g bda g C '@ g.b C /da C 'jr g b j dv g where g and g are the outward conormals with respect to the metric g: Observe that if b is not smooth on any part which is the K-level set of b, then on this is also the K-level set of b, which is smooth near this portion. Applying Sard s theorem, we can perturb K is piecewise C : Applying Sard s theorem again, we find a subseuence of positive going to 0; so that the are piecewise C : Then we show the above boundary integrals are nonnegative. The boundary integral portion is easily seen to be nonnegative, because either ' 0 g b g.b C / 0 there. The boundary integral portion in the interior of is also nonnegative, because there we have b b C.and b b C g g.b C g g.b C / 0: Taking the limit along the (Sard) seuence of going to 0; we obtain./! up to a set of measure zero, and hr g ';r g bi g dv g hr g ';r g bi g dv g jr g bj dv g jr g bj dv g : Case. S has positive measure. The discriminant. /. /. / is an analytic function in ; because the smooth u is actually analytic (cf. [7, p. 0]). So must vanish identically. Then we have either.x/.x/ or.x/.x/ at any point x : In turn, we know that.x/.x/.x/ tan. 6 / and b K > b.x/ at every boundary point of S inside \ V : If the boundary has positive measure, then.x/.x/.x/ tan. 6 / everywhere by the analyticity of u; and (.0) is trivially true. In the case has zero measure, b b >Kis smooth up to the boundary of every component of fx j b.x/ > Kg: By the pointwise Jacobi ineualities (.4) and (.8), the integral ineuality (.0) is also valid in case.

13 HESSIAN ESTIMATES 7 Proof of Theorem. We assume that R 4 and u is a solution on R for simplicity of notation. By scaling v.x/ u. R 4 x/=. R 4 / ; we still get the estimate in Theorem.. Without loss of generality, we assume that the continuous Hessian u sits on the convex branch of f. ; ; / j C C g containing.; ; /= p ; then u satisfies (.) with n and. By symmetry this also covers the concave branch corresponding to. Step. By the integral Jacobi ineuality (.0) in Proposition.4, b is subharmonic in the integral sense; then b is also subharmonic in the integral sense on the minimal surface M.x; u/: hr g ';r g b i g dv g hr g.b '/ 6b'r g b; r g bi g dv g.'b jr g bj C 6b'jr g bj /dv g 0 for all nonnegative ' C0, where we approximate b ' by smooth functions if necessary. Applying Michael and Simon s mean value ineuality [6, theorem.4] to the Lipschitz subharmonic function b, we obtain = = b.0/ C./ b dv g C./ b dv g ; B \M where B r is the ball with radius r and center.0; u.0// in R R, and B r is the ball with radius r and center 0 in R. Choose a cutoff function ' C0.B / such that ' 0, ' on B, and j'j :; we then have = = = b dv g ' 6 b dv g.'b = / 6 dv g : B B B Applying the Sobolev ineuality on the minimal surface M [6, theorem.] or [, theorem 7.] to 'b =, which we may assume to be C by approximation, we obtain =.'b = / 6 dv g C./ jr g.'b = /j dv g : B B Splitting the second integrand as follows: jr g.'b = /j ˇ b 'r gb C b = r = g ' ˇ B b ' jr g bj C bjr g 'j ' jr g bj C bjr g 'j ;

14 8 M. WARREN AN Y. YUAN where we used b, we get b.0/ C./ jr g.'b = /j dv g B C./ ' jr g bj dv g C bjr g 'j dv g B B C./kuk L.B / C C./Œkuk L ƒ.b / Ckuk L. / : ƒ Step Step Step. By (.0) in Proposition.4, b satisfies the Jacobi ineuality in the integral sense: 4 g b jr g bj : Multiplying both sides by the above nonnegative cutoff function ' C0.B / and then integrating, we obtain ' jr g bj dv g ' 4 g bdv g B B h'r g ';r g bidv g B ' jr g bj dv g C 8 jr g 'j dv g : B B It follows that ' jr g bj dv g 6 B B jr g 'j dv g : Observe the ( conformality ) identity: C ; C ; C V. ; ; / where we used the identity V Q i. C i / with. We then have (.) jr g 'j dv g X. i '/ X C Vdx. i '/. i /dx i i :4 4 udx: i

15 HESSIAN ESTIMATES 9 Thus ' jr g bj dv g C./ 4udx B B C./kuk L.B /: Step. By (.), we get bjr g 'j dv g C./ b 4 udx: B Choose another cutoff function C0 j j:. Wehave b 4 udx b 4 udx B.B / such that 0, on B, and hb C b; uidx Now B B B kuk L.B /.bj jc jbj/dx B C./kuk L.B /.b Cjbj/dx: B b max ln C max ;K max C K< C C C K 4uCK; where C >0follows from arctan C arctan arctan >0. Hence bdx C./ΠCkuk L.B / : B We have left to estimate R B jbjdx: v ux jbjdx t. i b/. C i /. C /. C /. C /dx B B i jr g bjv dx B B = jr g bj Vdx B = Vdx :

16 0 M. WARREN AN Y. YUAN Repeating the Jacobi argument from Step, we see jr g bj VdxC./kuk L. /: B Then by the Sobolev ineuality on the minimal surface M, wehave Vdx dv g 6 dv g C./ jr g j dv g ; B B where the nonnegative cutoff function C0./ satisfies on B and jj :. Applying the conformality euality (.) again, we obtain jr g j dv g C./ 4udx C./kuk L. /: Thus we get and B Vdx C./kuk L. / B jbjdx C./kuk L. / : In turn, we obtain bjr g 'j dv g C./ŒKkuk L.B / Ckuk L.B / Ckuk L. / : B Finally, collecting all the estimates in the above three steps, we arrive at max.0/ exp C./ kuk L. / Ckuk L. / Ckuk L. / C./ exp C./kuk L. / : This completes the proof of Theorem.. Remark. A sharper Hessian estimate and a gradient estimate for the special Lagrangian euation (.) with n were derived by an elementary method in []. More involved arguments are needed to obtain the Hessian and gradient estimates for (.) with n and j j > in [4].

17 HESSIAN ESTIMATES Bibliography [] Allard, W. K. On the first variation of a varifold. Ann. of Math. () 95 (97), [] Bao, J.; Chen, J. Optimal regularity for convex strong solutions of special Lagrangian euations in dimension. Indiana Univ. Math. J. 5 (00), no. 5, 49. [] Chou, K.-S.; Wang, X.-J. A variational theory of the Hessian euation. Comm. Pure Appl. Math. 54 (00), no. 9, [4] Harvey, R.; Lawson, H. B., Jr. Calibrated geometries. Acta Math. 48 (98), [5] Heinz, E. On elliptic Monge-Ampère euations and Weyl s embedding problem. J. Analyse Math. 7 (959), 5. [6] Michael, J. H.; Simon, L. M. Sobolev and mean-value ineualities on generalized submanifolds of R n. Comm. Pure Appl. Math. 6 (97), [7] Morrey, C. B., Jr. On the analyticity of the solutions of analytic non-linear elliptic systems of partial differential euations. I. Analyticity in the interior. Amer. J. Math. 80 (958), [8] Pogorelov, A. V. The Minkowski multidimensional problem. Scripta Series in Mathematics. Winston, Washington,.C.; Halsted [Wiley], New York Toronto London, 978. [9] Trudinger, N. S. Weak solutions of Hessian euations. Comm. Partial ifferential Euations (997), no. 7-8, 5 6. [0] Urbas, J. I. E. On the existence of nonclassical solutions for two classes of fully nonlinear elliptic euations. Indiana Univ. Math. J. 9 (990), no., [] Urbas, J. Some interior regularity results for solutions of Hessian euations. Calc. Var. Partial ifferential Euations (000), no.,. [] Urbas, J. An interior second derivative bound for solutions of Hessian euations. Calc. Var. Partial ifferential Euations (00), no. 4, [] Warren, M.; Yuan, Y. Explicit gradient estimates for minimal Lagrangian surfaces of dimension two. arxiv: , 007. [4] Warren, M.; Yuan, Y. Hessian and gradient estimates for three dimensional special Lagrangian euations with large phase. arxiv: [5] Yuan, Y. A Bernstein problem for special Lagrangian euations. Invent. Math. 50 (00), no., 7 5. [6] Yuan, Y. Global solutions to special Lagrangian euations. Proc. Amer. Math. Soc. 4 (006), no. 5, MICAH WARREN University of Washington epartment of Mathematics Box 5450 Seattle, WA mwarren@ math.washington.edu Received August 007. YU YUAN University of Washington epartment of Mathematics Box 5450 Seattle, WA yuan@ math.washington.edu