On a constant rank theorem  for nonlinear elliptic PDEs

Gábor  Székelyhidi; Ben  Weinkove

doi:10.3934/dcds.2016081

Article Contents

2016, Volume 36, Issue 11: 6523-6532. Doi: 10.3934/dcds.2016081

This issue Previous Article Weak KAM theory for HAMILTON-JACOBI equations depending on unknown functions Next Article Mixing invariant extremal distributional chaos

On a constant rank theorem for nonlinear elliptic PDEs

Gábor Székelyhidi^1, and
Ben Weinkove^2,

1.
Department of Mathematics, University of Notre Dame, 255 Hurley, Notre Dame, IN 46556
2.
Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208

Received: September 30, 2015

Revised: May 31, 2016

Published: May 2017

Abstract Related Papers Cited by

Abstract

We give a new proof of Bian-Guan's constant rank theorem for nonlinear elliptic equations. Our approach is to use a linear expression of the eigenvalues of the Hessian instead of quotients of elementary symmetric functions.

Keywords:

Mathematics Subject Classification: Primary: 35J60.

Citation:

Related Papers

Cited by

References

[1]	O. Alvarez, J.-M. Lasry and P.-L. Lions, Convex viscosity solutions and state constraints, J. Math. Pures Appl. (9), 76 (1997), 265-288.doi: 10.1016/S0021-7824(97)89952-7.
[2]	B. Bian and P. Guan, A microscopic convexity principle for nonlinear partial differential equations, Invent. Math., 177 (2009), 307-335.doi: 10.1007/s00222-009-0179-5.
[3]	B. Bian and P. Guan, A structural condition for microscopic convexity principle, Discrete Contin. Dyn. Syst., 28 (2010), 789-807.doi: 10.3934/dcds.2010.28.789.
[4]	H. J. Brascamp and E. H. Lieb, On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation, J. Functional Analysis, 22 (1976), 366-389.doi: 10.1016/0022-1236(76)90004-5.
[5]	L. Caffarelli and A. Friedman, Convexity of solutions of some semilinear elliptic equations, Duke Math. J., 52 (1985), 431-455.doi: 10.1215/S0012-7094-85-05221-4.
[6]	L. Caffarelli, P. Guan and X.-N. Ma, A constant rank theorem for solutions of fully nonlinear elliptic equations, Comm. Pure Appl. Math., 60 (2007), 1769-1791.doi: 10.1002/cpa.20197.
[7]	L. Caffarelli and J. Spruck, Convexity properties of solutions to some classical variational problems, Comm. Partial Differential Equations, 7 (1982), 1337-1379.doi: 10.1080/03605308208820254.
[8]	P. Cannarsa and C. Sinestrari, Semiconcave functions, Hamilton-Jacobi equations, and optimal control, in Progress in Nonlinear Differential Equations and their Applications 58, Birkhäuser Boston, Inc., Boston, MA, 2004.
[9]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2001.
[10]	P. Guan, Q. Li and X. Zhang, A uniqueness theorem in Kähler geometry, Math. Ann., 345 (2009), 377-393.doi: 10.1007/s00208-009-0358-0.
[11]	P. Guan, C. S. Lin and X.-N. Ma, The Christoffel-Minkowski problem II: Weingarten curvature equations, Chin. Ann. Math., Ser. B, 27 (2006), 595-614.doi: 10.1007/s11401-005-0575-0.
[12]	P. Guan and X.-N. Ma, The Christoffel-Minkowski problem I: Convexity of solutions of a Hessian equations, Invent. Math., 151 (2003), 553-577.doi: 10.1007/s00222-002-0259-2.
[13]	P. Guan, X.-N. Ma and F. Zhou, The Christoffel-Minkowski problem III: existence and convexity of admissible solutions, Comm. Pure Appl. Math., 59 (2006), 1352-1376.doi: 10.1002/cpa.20118.
[14]	P. Guan and D. H. Phong, A maximum rank problem for degenerate elliptic fully nonlinear equations, Math. Ann., 354 (2012), 147-169.doi: 10.1007/s00208-011-0729-1.
[15]	F. Han, X.-N. Ma and D. Wu, A constant rank theorem for Hermitian $k$-convex solutions of complex Laplace equations, Methods Appl. Anal., 16 (2009), 263-289.doi: 10.4310/MAA.2009.v16.n2.a5.
[16]	B. Kawohl, 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 (1986), 93-101.doi: 10.1002/mma.1670080107.
[17]	A. U. Kennington, Power concavity and boundary value problems, Indiana Univ. Math. J., 34 (1985), 687-704.doi: 10.1512/iumj.1985.34.34036.
[18]	N. J. Korevaar, Capillary surface convexity above convex domains, Indiana Univ. Math. J., 32 (1983), 73-81.doi: 10.1512/iumj.1983.32.32007.
[19]	N. J. Korevaar and J. L. Lewis, Convex solutions of certain elliptic equations have constant rank Hessians, Arch. Rational Mech. Anal., 97 (1987), 19-32.doi: 10.1007/BF00279844.
[20]	Q. Li, Constant rank theorem in complex variables, Indiana Univ. Math. J., 58 (2009), 1235-1256.doi: 10.1512/iumj.2009.58.3574.
[21]	X.-N. Ma and L. Xu, The convexity of solutions of a class of Hessian equation in bounded convex domain in $\mathbbR^3$, J. Funct. Anal., 255 (2008), 1713-1723.doi: 10.1016/j.jfa.2008.06.008.
[22]	I. Singer, I. B. Wong, S.-T. Yau and S.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 (1985), 319-333.
[23]	J. Spruck, Geometric aspects of the theory of fully nonlinear elliptic equations in Global Theory of Minimal Surfaces, Amer. Math. Soc., Providence, RI, (2005), 283-309.
[24]	G. Székelyhidi, Fully non-linear elliptic equations on compact Hermitian manifolds, preprint, arXiv:1501.02762.
[25]	G. Székelyhidi, V. Tosatti and B. Weinkove, Gauduchon metrics with prescribed volume form, preprint, arXiv:1503.04491.
[26]	N. S. Trudinger, Comparison principles and pointwise estimates for viscosity solutions of nonlinear elliptic equations, Revista Mat. Iber., 4 (1988), 453-468.doi: 10.4171/RMI/80.
[27]	X. J. Wang, Counterexample to the convexity of level sets of solutions to the mean curvature equation, J. Eur. Math. Soc., 16 (2014), 1173-1182.doi: 10.4171/JEMS/457.
[28]	M. Warren and Y. Yuan, Hessian estimates for the sigma-2 equation in dimension 3, Comm. Pure Appl. Math., 62 (2009), 305-321.doi: 10.1002/cpa.20251.