Abstract
We study microscopic convexity property of fully nonlinear elliptic and parabolic partial differential equations. Under certain general structure condition, we establish that the rank of Hessian ∇ 2 u is of constant rank for any convex solution u of equation F(∇ 2 u,∇ u,u,x)=0. The similar result is also proved for parabolic equations. Some of geometric applications are also discussed.
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Alexandrov, A.D.: Zur Theorie der gemischten Volumina von konvexen körpern, III. Die Erweiterung zweier Lehrsätze Minkowskis uber die konvexen polyeder auf beliebige konvexe Flächen. Mat. Sb. N.S. 3, 27–46 (1938) (in Russian)
Alexandrov, A.V.: Über konvexe Flächen mit ebenen Schattengrenzen. Rec. Math. N.S. [Mat. Sb.] 5(47), 309–316 (1939) (in Russian)
Alexandrov, A.D.: Uniqueness theorems for surfaces in the large. I. Vestn. Leningr. Univ. 11, 5–17 (1956) (in Russian). English translation: AMS Transl., Ser. 2, 21, 341–354 (1962)
Alvarez, O., Lasry, J.M., Lions, P.-L.: Convexity viscosity solutions and state constraints. J. Math. Pures Appl. 76, 265–288 (1997)
Andrews, B.: Pinching estimates and motion of hypersurfaces by curvature functions. J. Reine Angew. Math. 608, 17–33 (2007)
Brascamp, H.J., Lieb, E.H.: On extensions of the Brunn-Minkowski and Prekopa-Leindler theorems, including inequalities for log-concave functions, with an application to the diffusion equation. J. Funct. Anal. 22, 366–389 (1976)
Caffarelli, L., Friedman, A.: Convexity of solutions of some semilinear elliptic equations. Duke Math. J. 52, 431–455 (1985)
Caffarelli, L., Spruck, J.: Convexity properties of solutions to some classical variational problems. Commun. Partial Differ. Equ. 7, 1337–1379 (1982)
Caffarelli, L., Guan, P., Ma, X.: A constant rank theorem for solutions of fully nonlinear elliptic equations. Commun. Pure Appl. Math. 60, 1769–1791 (2007)
Caffarelli, L., Nirenberg, L., Spruck, J.: The Dirichlet problem for nonlinear second order elliptic equations, III: Functions of the eigenvalues of the Hessian. Acta Math. 155, 261–301 (1985)
Cheng, S.Y., Yau, S.T.: Hypersurfaces with constant scalar curvature. Math. Ann. 225, 195–204 (1977)
Chern, S.S.: Some new characterizations of the Euclidean sphere. Duke Math. J. 12, 279–290 (1945)
Ecker, K., Huisken, G.: Immersed hypersurfaces with constant Weingarten curvature. Math. Ann. 283, 329–332 (1989)
Guan, P.: C 2 A priori estimates for degenerate Monge-Ampere equations. Duke Math. J. 86, 323–346 (1997)
Guan, P., Ma, X.N.: The Christoffel-Minkowski problem I: Convexity of solutions of a Hessian equations. Invent. Math. 151, 553–577 (2003)
Guan, P., Lin, C.S., Ma, X.N.: The Christoffel-Minkowski problem II: Weingarten curvature equations. Chin. Ann. Math., Ser. B 27, 595–614 (2006)
Guan, P., Ma, X.N., Zhou, F.: The Christoffel-Minkowski problem III: existence and convexity of admissible solutions. Commun. Pure Appl. Math. 59, 1352–1376 (2006)
Guan, P., Li, Q., Zhang, X.: A uniqueness theorem in Kähler geometry. Preprint (2007)
Hamilton, R.S.: Four manifolds with positive curvature operator. J. Differ. Geom. 24, 153–179 (1986)
Hartman, P., Nirenberg, L.: On spherical image maps whose Jacobians do not change sign. Am. J. Math. 81, 901–920 (1959)
Huisken, G.: Flow by mean curvature of convex surfaces into spheres. J. Differ. Geom. 20, 237–266 (1984)
Huisken, G., Sinestrari, C.: Convexity estimates for mean curvature flow and singularities of mean convex surfaces. Acta Math. 183, 45–70 (1999)
Kawohl, B.: A remark on N. Korevaar’s concavity maximum principle and on the asymptotic uniqueness of solutions to the plasma problem. Math. Methods Appl. Sci. 8, 93–101 (1986)
Kennington, A.U.: Power concavity and boundary value problems. Indiana Univ. Math. J. 34, 687–704 (1985)
Korevaar, N.J.: Capillary surface convexity above convex domains. Indiana Univ. Math. J. 32, 73–81 (1983)
Korevaar, N.J.: Convex solutions to nonlinear elliptic and parabolic boundary value problems. Indiana Univ. Math. J. 32, 603–614 (1983)
Korevaar, N.J., Lewis, J.: Convex solutions of certain elliptic equations have constant rank Hessians. Arch. Ration. Mech. Anal. 91, 19–32 (1987)
Singer, I., Wong, B., Yau, S.T.: Stephen S.T. Yau, An estimate of gap of the first two eigenvalues in the Schrodinger operator. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 12, 319–333 (1985)
Treves, F.: A new method proof of the subelliptic estimates. Commun. Pure Appl. Math. 24, 71–115 (1971)
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Research of the first author was supported in part by NSFC No.10671144 and National Basic Research Program of China (2007CB814903). Research of the second author was supported in part by an NSERC Discovery Grant.
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Bian, B., Guan, P. A microscopic convexity principle for nonlinear partial differential equations. Invent. math. 177, 307–335 (2009). https://doi.org/10.1007/s00222-009-0179-5
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DOI: https://doi.org/10.1007/s00222-009-0179-5