THE INTERIOR C 2 ESTIMATE FOR PRESCRIBED GAUSS CURVATURE EQUATION IN DIMENSION TWO

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1 THE INTERIOR C ESTIMATE FOR PRESCRIBED GAUSS CURVATURE EQUATION IN DIMENSION TWO CHUANQIANG CHEN, FEI HAN, AND QIANZHONG OU arxiv:5.047v [math.ap 8 Apr 06 Abstract. In this paper, we introduce a new auxiliary function, and establish the interior C estimate for prescribed Gauss curvature equation in dimension two.. Introduction Given a positive function f(x) C (Ω) with Ω R n, to find a convex solution u, such that the Gauss curvature of the graph (x,u(x)) is f(x), that is (.) det u (+ u ) n+ f(x), in Ω. This is the classical prescribed Gauss curvature problem, and (.) is called prescribed Gauss curvature equation. The a priori estimates are very important for fully nonlinear elliptic equations, especially the C estimate. The interior C estimate of σ Hessian equation (.) σ ( u) f(x), in B R (0) R n in higher dimensions is a longstanding problem, where σ ( u) σ (λ( u)) λ i λ i, λ( u) (λ,,λ n ) are the eigenvalues of u, and f > 0. For i <i n n, (.) is the Monge-Ampère equation, and Heinz [5 obtained the estimate by the convex hypersurface geometry method (see [ for an elementary analytic proof). For Monge-Ampère equations with dimension n 3, Pogorelov [7 constructed his 00 Mathematics Subject Classification: 35J96, 35B45, 35B65. Keywords: interiorc estimate, Monge-Ampèreequation, prescribed Gauss curvature equation. Research of the first author is supported by the National Natural Science Foundation of China (NO and NO. 4788). Research of the second author is supported by the National Natural Science Foundation of China (NO. 6048). Research of the third author is supported by NSFC No and by Guangxi Science Foundation (04GXNSFAA808) and Guangxi Education Department Key Laboratory of Symbolic Computation and Engineering Data Processing.

2 CHUANQIANG CHEN, FEI HAN, AND QIANZHONG OU famous counterexample, namely irregular solutions to Monge-Ampère equations. For n 3 and f, (.) can be reduced to a special Lagrangian equation after a Lewy transformation, and Warren-Yuan obtained the corresponding interior C estimate in the celebrated paper [0. Moreover, the problem is still open for general f with n 4 and nonconstant f with n 3. Also Urbas [9 generalized the counterexample for σ k Hessian equations with k 3. Moreover, Pogorelov type estimates for the Monge-Ampère equations and σ k Hessian equation (k ) were derived by Pogorelov [7 and Chou-Wang [3 respectively, and see [4 and [6 for some generalizations. In this paper, we consider the convex solution of prescribed Gauss curvature equation in dimension n as follows (.3) det u (+ u ) f(x), in B R(0) R, and establish the interior C estimate as follows Theorem.. Suppose u C 4 (B R (0)) be a convex solution of prescribed Gauss curvature equation (.3) in dimension n, where 0 < m f M in B R (0). Then (.4) u(0) C(m,M,R,sup f,sup f,sup u ), where C is a positive constant depending only on m, M, R, sup f, sup f, and sup u. Remark.. Heinz established an interior C estimate for general Monge-Ampère equation with general conditions in [5, and the proof depends on the strict convexity of solutions and the geometry of convex hypersurface in dimension two. In this paper, our proof, which is based on a suitable choice of auxiliary functions, is elementary and avoids geometric computations on the graph of solutions. This technique is from [. Remark.3. By Trudinger s gradient estimates of Hessian equations in [8 or a gradient estimate of convex function, that is osc u B R (0) sup u B R (0) R, we can bound u(0) in terms of u.

3 C INTERIOR A PRIORI ESTIMATE 3 The rest of the paper is organized as follows. In Section, we give the calculations of the derivatives of eigenvalues and eigenvectors with respect to the matrix. In Section 3, we introduce a new auxiliary function, and prove Theorem... Derivatives of eigenvalues and eigenvectors In this section, we give the calculations of the derivatives of eigenvalues and eigenvectors with respect to the matrix. We think the following result is known for many people, for example see [ for a similar result. For completeness, we give the result and a detailed proof. Proposition.. Let W {W ij } is an n n symmetric matrix and λ(w) (λ,λ,,λ n ) are the eigenvalues of the symmetric matrix W, and the corresponding continuous eigenvector field is τ i (τ i,,,τ i,n) S n. Suppose that W {W ij } is diagonal, λ i W ii and the corresponding eigenvector τ i (0,,0,,0,,0) S n at the diagonal matrix W. If λ k is distinct with other i th eigenvalues, then we have at the diagonal matrix W (.) (.) (.3) (.4) τ k,k 0, p,q; W ik τ k,k W pk W pk (λ k λ p ), p k; τ k,i W ik W ii τ k,i W iq W qk, i k; ( ), i k; τ k,i λ k λ q, i k,i q,q k; 0, otherwise. W ik W kk ( ), i k; (.5) τ k,i W rs 0, otherwise. Proof. From the definition of eigenvalue and eigenvector of matrix W, we have (W λ k I)τ k 0, where τ k is the eigenvector of W corresponding to the eigenvalue λ k. That is, for i,,n, it holds (.6) [W ii λ k τ k,i + j i W ij τ k,j 0.

4 4 CHUANQIANG CHEN, FEI HAN, AND QIANZHONG OU When W {W ij } is diagonal and λ k is distinct with other eigenvalues, λ k and τ k are C at the matrix W. In fact, (.7) τ k,k, τ k,i 0, i k, at W. Taking the first derivative of (.6), we have [ W ii λ k τ k,i +[W ii λ k τk,i + [ W ij τ k τ k,j,j +W ij 0. j i Hence for i k, we get from (.7) (.8) And for i k, then (.9) Since τ k S n, we have (.0) λ k W kk [W ii λ k τk,i + j i {, p k,q k; 0, otherwise. W ij τ k,j 0, { W ik λ k λ i, p i,q k; 0, otherwise. τ k (τ k,) + +(τ k,k) + +(τ k,n). Taking the first derivative of (.0), and using (.7), it holds (.) τ k,k 0, (p,q). For i k, taking the second derivative of (.6), and using (.7), it holds hence W kk [ λ k τ k,k + W rs W rs j k [ W kj τ k,j W rs + W kj W rs τ k,j 0, (.) λ k W rs j k 0, otherwise. [ W kj τ k,j W rs + W kj W rs τ k,j λ k λ q, p k,q k,r q,s k; λ k λ s, r k,s k,p s,q k;

5 C INTERIOR A PRIORI ESTIMATE 5 For i k, it holds [ W ii λ k τk,i +[ W ii λ k τk,i τ k,i +[W ii λ k W rs W rs W rs W rs + j i [ W ij τ k,j W rs + W ij W rs τ k,j 0, then (.3) (.4) (.5) τ k,i W ik W ii τ k,i W iq W qk τ k,i W ik W kk W ik W iq W iq τ k,q W qk, i k; λ k λ q, i k,i q,q k; [ λ k W kk W ik, i k; (.6) then τ k,i W rs 0, otherwise. From (.0), we have τ k,k W rs i k τ,k k τ k,k + W rs i k W rs W rs 0, { λ k λ p λ k λ p, p k,q k,r p,s q; 0, otherwise. The proof of Proposition. is finished. Example.. When n, the matrix ( ) u u has two eigenvalues u u λ ( +u )+ ( u ) +4u u, λ ( +u ) ( u ) +4u u, with λ λ. If λ > λ, [( ) ( u u 0 λ u u 0 )( ) ξ 0, ξ

6 6 CHUANQIANG CHEN, FEI HAN, AND QIANZHONG OU we can get ξ (u ) ( u ) +4u u ; ξ u. Then the eigenvector τ corresponding to λ is We can verify Proposition.. τ (ξ,ξ ) ξ +ξ. 3. Proof of Theorem. Now we start to prove Theorem.. Letτ(x) τ( u(x)) (τ,τ ) S bethecontinuouseigenvector fieldof u(x) corresponding to the largest eigenvalue. Denote (3.) Σ : {x B R (0) : r x + x,τ(x) > 0,r x,τ(x) > 0}, where r R. It is easy to know, Σ is an open set and B r (0) Σ B R (0). We introduce a new auxiliary function in Σ as follows (3.) φ(x) (x) β g( Du )u ττ where (x) (r x + x,τ(x) )(r x,τ(x) ) with β 4 and g(t) e c 0 r t with c 0 3. In fact, x,τ(x) is invariant under rotations of the coordinates, so is (x). m From the definition of Σ, we know (x) > 0 in Σ, and 0 on Σ. Assume the maximum of φ(x) in Σ is attains at x 0 Σ. By rotating the coordinates, we can assume u(x 0 ) is diagonal. In the following, we denote λ i u ii (x 0 ), λ (λ,λ ). Without loss of generality, we can assume λ λ. Then τ(x 0 ) (,0). We will use notion h O(f) if h(x) Cf(x) for any x Ω with a positive constantc dependingonlyonm, M, R, sup f,sup f,andsup u. Similarly we write h O(f) if h(x) Cf(x) and h O(f) if h(x) Cf(x). Now, we assume λ is big enough. Otherwise there is nothing to prove. Then we have from the equation (.3), (3.3) λ f(+ u ) λ M(+ u ) λ < λ.

7 C INTERIOR A PRIORI ESTIMATE 7 Hence λ is distinct with the other eigenvalue, and τ(x) is C at x 0. Moreover, the test function (3.4) ϕ βlog +logg( u )+log attains the local maximum at x 0. In the following, all the calculations are at x 0. Then, we can get 0 ϕ i β i + g u k u ki + i, g so we have k (3.5) i β i g g u iu ii, i,. At x 0, we also have since g g g 0. Let 0 ϕ ii β[ ii i g g g + g + g g k u k u ki u l u li (u ki u ki +u k u kii )+ ii u i u β[ ii i g + g [u ii + k k l u k u kii + ii u i, u F det u F det u u λ, F det u u λ, 0, F det u u 0. Then from the equation (.3) we can get (3.6) λ f(+ u ) λ. Differentiating (.3) once, we can get (3.7) F i +F u i f i (+ u ) +f (+ u ) u i u ii,

8 8 CHUANQIANG CHEN, FEI HAN, AND QIANZHONG OU then (3.8) u i F [f i(+ u ) +f (+ u ) u i u ii F i f i(+ u ) λ f(+ u ) λ +f 4(+ u u ii )u i f(+ u ) λ λ i +O(). i Differentiating (.3) twice, we can get F +F u [f(+ u ) det u u x u det u u u u u f (+ u ) +f (+ u ) u +f[(u ) +(+ u ) (u +u ku k ) u +u f (+ u ) +8f (+ u )u +f[8u +4(+ u )u +4f(+ u )[u +u u +u [ f (+ u ) +f 4(+ u )u f(+ u ) λ λ (3.9) f (+ u ) +8f (+ u )u +f[8u +4(+ u )u +4f(+ u )[ u +u u f (+ u ) +u +f(+ u ) ( ) f[8u +4(+ u )u +O(λ ) +4f(+ u )[ u +u f (+ u ) +u +f(+ u ) ( ),

9 C INTERIOR A PRIORI ESTIMATE 9 and F +F u x x [f(+ u ) det u u det u u det u u u u u u u f (+ u ) +f (+ u ) u u +f (+ u ) u +f[(u )(u u )+(+ u ) u k u k u u + u f (+ u ) +4(+ u )[f u u +f u (3.0) +8f u u (+ u ) +4f(+ u )u +4f(+ u )u [ f(+ u ) λ +O() [ f (+ u ) λ +f 4(+ u )u u λ f(+ u ) λ + [ f (+ u ) λ +f 4(+ u )u f(+ u ) λ [f (+ u ) +f 8(+ u )u u + [f (+ u ) +f 8(+ u )u +O(λ ).

10 0 CHUANQIANG CHEN, FEI HAN, AND QIANZHONG OU Hence we can get by (3.7) and (3.9), 0 F ii ϕ ii i β F ii [ ii i g + g i i + i F ii ii i F ii u ii + g g F ii [ i u k F ii u iik βλ [ +βλ [ g + g [λ +λ f(+ u ) + g g [(u f +u f )(+ u ) +4f (+ u )(u +u u ) + { f[8u +4(+ u )u +O(λ ) +4f(+ u )[ u +u u f (+ u ) +u +f(+ u ) ( ) } λ [ λ [ β[λ +λ βλ βλ + g g f(+ u ) λ +λ [ [4f(+ u )u + f (+ u ) k i (3.) +f[8u +4(+ u ) + λ [ + λ [β + g g u u +4f(+ u )u +O() β[λ +λ βλ + g g f(+ u ) λ + λ [ + λ [ +( β β)λ +βf(+ u )[(+ u ) g g 4 u +O(),

11 where we used the following inequalities C INTERIOR A PRIORI ESTIMATE λ [ [4f(+ u )u + f (+ u ) +f[8u u +4(+ u ) λ [ + λ { λ [4f(+ u )u + f (+ u ) } [4f(+ u )u + f (+ u ) +f[8u λ u +4(+ u ) λ [ λ )u f(+ u ) [4f(+ u + f (+ u ) +f[8u +4(+ u ) λ [ 8fu λ 8f (+ u )u f (+ u ) f +f[8u +4(+ u ) λ [ 8f (+ u )u f (+ u ), f and 4f(+ u )u 4f(+ u )u [ β g g u u 4βf(+ u ) u +O(). Now we have the following lemma. Lemma 3.. If λ is big enough, we have at x 0 (3.) (3.3) and (3.4) βλ λ 4 ( ) +O( ); βf(+ u )[(+ u ) g g 4 u O( ); β[λ +λ g g fλ βλ [ λ ( ) λ 4 ( ) +O( ). Proof. At x 0, τ (τ,τ ) (,0). Then from Proposition. we get τ m τ m τ x, i τ x m x m u pqi x u pqi x i u pq u pq (3.5) m m x u i λ λ, i,.

12 CHUANQIANG CHEN, FEI HAN, AND QIANZHONG OU From the definition of, then we have at x 0 (3.6) [r x + x,τ [r x,τ (r x )(r x ). Taking the first derivative of, we can get i [ x i + x,τ x,τ i [r x,τ +[r x + x,τ [ x,τ x,τ i [ x i +x (δ i + x, i τ )[r x +(r x )[ x (δ i + x, i τ ) { x (r x )+x x, τ (x x ), i ; x (r x )+x x, τ (x x ), i. Hence if λ is big enough, we can get (3.7) Also we have βλ βλ [ x (r x ) x +x x x λ λ βλ [ 8r6 + 8r8 ( λ λ ) λ 4 ( ) +O( ). x (r x ) x x +x x x (r x ) x (r x ) +x x x x +x x x x u λ λ [ f(+ u ) +O() λ λ λ (x x [ x )(r x ) O() λ λ f(+ u ) [β λ λ λ + g g u x (r x ) [+O( (x ) x x )(r x ) λ +x x x x x (r x ) [+O( ) λ f(+ u ) g λ λ g u f(+ u ) β[ x (r x ) x x +x x λ λ λ x x +[x x f(+ u ) β[ β (λ λ ) g g u u, λ λ

13 C INTERIOR A PRIORI ESTIMATE 3 then we can get [+β x x (x x ) f(+ u ) (λ λ ) x (r x ) [+O( x x ) [x x λ x (r x ) [+O( )+x λ x (x x ) (r x ) (3.8) x (r x ) [+O( ), λ which implies f(+ u ) (λ λ ) β g g u u f(+ u ) (λ λ ) β g g u u (3.9) x (r x ) [+O( ). λ That is x (r x ) (3.0) so if λ is big enough. Hence βf(+ u )[(+ u ) g g 4 u O( ). Taking second derivatives of, we can get ii [ + x,τ x,τ ii + x,τ i x,τ i [r x,τ +[ x i + x,τ x,τ i [ x,τ x,τ i +[r x + x,τ [ x,τ x,τ ii x,τ i x,τ i [ +x x,τ ii +(δ i + x, i τ ) [r x +[ x i +x (δ i + x, i τ )[ x (δ i + x, i τ ) +(r x )[ x x,τ ii (δ i + x, i τ ), (3.) (3.) (r x ) x (x x ) x,τ +(4x x ) x, τ +(x 0x ) x, τ, (r x ) x (x x ) x,τ +8x x x, τ +(x 0x ) x, τ.

14 4 CHUANQIANG CHEN, FEI HAN, AND QIANZHONG OU Hence β[λ +λ β[λ r x (3.3) r x +λ β x (x x ) [λ x,τ +λ x,τ +βλ [ x (4x x ) + x (x 0x ) ( ) λ λ λ λ +βλ [ 8x x u + x (x 0x ) u ( ). λ λ λ λ Direct calculations yield x,τ [ x m τ u pq u pq + x m τ m τ x + m m x m τ m x x m [ τ m u pq u pq + τ m u pq u rs u pq u rs τ 0+x u pq u rs +x [ τ u pq + τ u pq u rs u pq [ +x x [ u λ λ λ λ u pq u rs u pq u rs [ u +x (λ λ ) + u (λ λ ), Similarly, we have x,τ [ x m τ u pq + u pq m [ λ λ [ +x λ λ x m τ m τ x + m x m τ m x x m [ τ m u pq u pq + τ m [ u u x λ λ u pq u rs u pq u rs u +x [ u (λ λ ) + u u (λ λ ),

15 C INTERIOR A PRIORI ESTIMATE 5 then λ x,τ +λ x,τ [ [ u u x λ +λ λ λ λ λ [ +x λ λ [ u x λ λ λ [ [f (+ u ) +f 8(+ Du )u u + [f (+ u ) +f 8(+ u )u +O(λ ) x (λ λ ) [ f (+ u ) +f (+ u ) u u [ +x f (+ u ) +f (+ u ) u (λ λ ) u [ u O( [ u )+ O( )+O() λ λ λ λ λ λ [ [ u u + O(λ )+ O(λ )+ (3.4) O( ). λ λ λ λ λ From (3.3) and (3.4), we can get β[λ +λ O( r x ) βλ +O( λ ) +( ) O( )+( ) O( )+ O( ) λ λ λ λ λ λ λ u +( ) O( λ λ λ )+ u O( λ λ λ ) βλ r x +O( )+( ) O( )+( λ λ λ ) O( λ λ + O( λ )+[ f(+ u ) λ +O() O( λ ) +[ f(+ u ) λ +O() O( ) λ ) (3.5) βλ r x +O( ) λ ( ) λ 4 ( ).

16 6 CHUANQIANG CHEN, FEI HAN, AND QIANZHONG OU r Now we just need to estimate βλ x. If x r, we can get βλ r x If x > r, we can get βλ r x if λ is big enough. Hence (3.6) and (3.7) 8 λ r x 6 r λ 8 λ r x x r x βλ [, βλ r x 8 r x c 0 r fλ g g fλ βλ [, g g fλ. λ βλ [ x r x β[λ +λ g g fλ βλ [ λ ( ) λ 4 ( ) +O( ). Now we continue to prove Theorem.. From (3.) and Lemma 3., we can get 0 F ii ϕ ii g g fλ +O( )+O() (3.8) So we can get i g g fλ C 0 C 0, (3.9) λ C. where C 0 and C are positive constants depending only on m, M, R, sup f, sup f, and sup u. So we can easily get and u τ(0)τ(0) (0) r 4βφ(0) r 4βφ(x 0) C, (3.30) u ξξ (0) u τ(0)τ(0) (0) C, ξ S. Then we have proved (.4), and Theorem. holds. Acknowledgement. The authors would like to express sincere gratitude to Prof. Xi-Nan Ma for the constant encouragement in this subject.

17 C INTERIOR A PRIORI ESTIMATE 7 References [ B. Andrews, Pinching estimates and motion of hypersurfaces by curvature functions, J. Reine Angew. Math., 608(007), [ C.Q. Chen, F. Han, Q.Z. Ou, The interior C estimate for Monge-Ampère equation in dimension n, arxiv:5.046, 05. [3 K.S. Chou, X.J. Wang, A variation theory of the Hessian equation, Comm. Pure Appl. Math., 54(9)(00), [4 P. Guan, C. Ren, Z. Wang, Global C estimates for convex solutions of curvature equations, Comm. Pure Appl. Math, 68(05), [5 E. Heinz, On elliptic Monge-Ampère equations and Weyls embedding problem, J. Analyse Math., 7(959), -5. [6 M. Li, C. Ren, Z. Wang, An interior estimate for convex solutions and a rigidity theorem, J. Funct. Anal., 70(06), [7 A.V. Pogorelov, The multidimensional Minkowski problem, New York, Wiley 978. [8 N.S. Trudinger, Weak solutions of Hessian equations, Comm. Partial Differential Equations, (7-8)(997), 5-6. [9 J. Urbas, On the existence of nonclassical solutions for two class of fully nonlinear elliptic equations, Indiana Univ. Math. J., 39(990), [0 M. Warren, Y. Yuan, Hessian estimates for the σ equation in dimension 3, Comm. Pure Appl. Math., 6(009), Department of Applied Mathematics,, Zhejiang University of Technology, Hangzhou, 3003, Zhejiang Province, CHINA address: cqchen@mail.ustc.edu.cn School of Mathematics Sciences, Xinjiang Normal University, Urumqi, , Xinjiang Uygur Autonomous Region, CHINA address: klhanqingfei@6.com Department of Mathematics, Hezhou University, Hezhou, 54800, Guangxi Province, CHINA address: ouqzh@63.com