# American Institute of Mathematical Sciences

November  2016, 36(11): 6523-6532. doi: 10.3934/dcds.2016081

## On a constant rank theorem for nonlinear elliptic PDEs

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

Received  October 2015 Revised  June 2016 Published  August 2016

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.
Citation: Gábor Székelyhidi, Ben Weinkove. On a constant rank theorem for nonlinear elliptic PDEs. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6523-6532. doi: 10.3934/dcds.2016081
##### 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. doi: 10.1016/S0021-7824(97)89952-7. Google Scholar [2] B. Bian and P. Guan, A microscopic convexity principle for nonlinear partial differential equations,, Invent. Math., 177 (2009), 307. doi: 10.1007/s00222-009-0179-5. Google Scholar [3] B. Bian and P. Guan, A structural condition for microscopic convexity principle,, Discrete Contin. Dyn. Syst., 28 (2010), 789. doi: 10.3934/dcds.2010.28.789. Google Scholar [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. doi: 10.1016/0022-1236(76)90004-5. Google Scholar [5] L. Caffarelli and A. Friedman, Convexity of solutions of some semilinear elliptic equations,, Duke Math. J., 52 (1985), 431. doi: 10.1215/S0012-7094-85-05221-4. Google Scholar [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. doi: 10.1002/cpa.20197. Google Scholar [7] L. Caffarelli and J. Spruck, Convexity properties of solutions to some classical variational problems,, Comm. Partial Differential Equations, 7 (1982), 1337. doi: 10.1080/03605308208820254. Google Scholar [8] P. Cannarsa and C. Sinestrari, Semiconcave functions, Hamilton-Jacobi equations, and optimal control,, in Progress in Nonlinear Differential Equations and their Applications 58, 58 (2004). Google Scholar [9] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Reprint of the 1998 edition. Classics in Mathematics. Springer-Verlag, (1998). Google Scholar [10] P. Guan, Q. Li and X. Zhang, A uniqueness theorem in Kähler geometry,, Math. Ann., 345 (2009), 377. doi: 10.1007/s00208-009-0358-0. Google Scholar [11] P. Guan, C. S. Lin and X.-N. Ma, The Christoffel-Minkowski problem II: Weingarten curvature equations,, Chin. Ann. Math., 27 (2006), 595. doi: 10.1007/s11401-005-0575-0. Google Scholar [12] P. Guan and X.-N. Ma, The Christoffel-Minkowski problem I: Convexity of solutions of a Hessian equations,, Invent. Math., 151 (2003), 553. doi: 10.1007/s00222-002-0259-2. Google Scholar [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. doi: 10.1002/cpa.20118. Google Scholar [14] P. Guan and D. H. Phong, A maximum rank problem for degenerate elliptic fully nonlinear equations,, Math. Ann., 354 (2012), 147. doi: 10.1007/s00208-011-0729-1. Google Scholar [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. doi: 10.4310/MAA.2009.v16.n2.a5. Google Scholar [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. doi: 10.1002/mma.1670080107. Google Scholar [17] A. U. Kennington, Power concavity and boundary value problems,, Indiana Univ. Math. J., 34 (1985), 687. doi: 10.1512/iumj.1985.34.34036. Google Scholar [18] N. J. Korevaar, Capillary surface convexity above convex domains,, Indiana Univ. Math. J., 32 (1983), 73. doi: 10.1512/iumj.1983.32.32007. Google Scholar [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. doi: 10.1007/BF00279844. Google Scholar [20] Q. Li, Constant rank theorem in complex variables,, Indiana Univ. Math. J., 58 (2009), 1235. doi: 10.1512/iumj.2009.58.3574. Google Scholar [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. doi: 10.1016/j.jfa.2008.06.008. Google Scholar [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. Google Scholar [23] J. Spruck, Geometric aspects of the theory of fully nonlinear elliptic equations, in Global Theory of Minimal Surfaces, (2005), 283. Google Scholar [24] G. Székelyhidi, Fully non-linear elliptic equations on compact Hermitian manifolds,, preprint, (). Google Scholar [25] G. Székelyhidi, V. Tosatti and B. Weinkove, Gauduchon metrics with prescribed volume form,, preprint, (). Google Scholar [26] N. S. Trudinger, Comparison principles and pointwise estimates for viscosity solutions of nonlinear elliptic equations,, Revista Mat. Iber., 4 (1988), 453. doi: 10.4171/RMI/80. Google Scholar [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. doi: 10.4171/JEMS/457. Google Scholar [28] M. Warren and Y. Yuan, Hessian estimates for the sigma-2 equation in dimension 3,, Comm. Pure Appl. Math., 62 (2009), 305. doi: 10.1002/cpa.20251. Google Scholar

show all references

##### 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. doi: 10.1016/S0021-7824(97)89952-7. Google Scholar [2] B. Bian and P. Guan, A microscopic convexity principle for nonlinear partial differential equations,, Invent. Math., 177 (2009), 307. doi: 10.1007/s00222-009-0179-5. Google Scholar [3] B. Bian and P. Guan, A structural condition for microscopic convexity principle,, Discrete Contin. Dyn. Syst., 28 (2010), 789. doi: 10.3934/dcds.2010.28.789. Google Scholar [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. doi: 10.1016/0022-1236(76)90004-5. Google Scholar [5] L. Caffarelli and A. Friedman, Convexity of solutions of some semilinear elliptic equations,, Duke Math. J., 52 (1985), 431. doi: 10.1215/S0012-7094-85-05221-4. Google Scholar [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. doi: 10.1002/cpa.20197. Google Scholar [7] L. Caffarelli and J. Spruck, Convexity properties of solutions to some classical variational problems,, Comm. Partial Differential Equations, 7 (1982), 1337. doi: 10.1080/03605308208820254. Google Scholar [8] P. Cannarsa and C. Sinestrari, Semiconcave functions, Hamilton-Jacobi equations, and optimal control,, in Progress in Nonlinear Differential Equations and their Applications 58, 58 (2004). Google Scholar [9] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Reprint of the 1998 edition. Classics in Mathematics. Springer-Verlag, (1998). Google Scholar [10] P. Guan, Q. Li and X. Zhang, A uniqueness theorem in Kähler geometry,, Math. Ann., 345 (2009), 377. doi: 10.1007/s00208-009-0358-0. Google Scholar [11] P. Guan, C. S. Lin and X.-N. Ma, The Christoffel-Minkowski problem II: Weingarten curvature equations,, Chin. Ann. Math., 27 (2006), 595. doi: 10.1007/s11401-005-0575-0. Google Scholar [12] P. Guan and X.-N. Ma, The Christoffel-Minkowski problem I: Convexity of solutions of a Hessian equations,, Invent. Math., 151 (2003), 553. doi: 10.1007/s00222-002-0259-2. Google Scholar [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. doi: 10.1002/cpa.20118. Google Scholar [14] P. Guan and D. H. Phong, A maximum rank problem for degenerate elliptic fully nonlinear equations,, Math. Ann., 354 (2012), 147. doi: 10.1007/s00208-011-0729-1. Google Scholar [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. doi: 10.4310/MAA.2009.v16.n2.a5. Google Scholar [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. doi: 10.1002/mma.1670080107. Google Scholar [17] A. U. Kennington, Power concavity and boundary value problems,, Indiana Univ. Math. J., 34 (1985), 687. doi: 10.1512/iumj.1985.34.34036. Google Scholar [18] N. J. Korevaar, Capillary surface convexity above convex domains,, Indiana Univ. Math. J., 32 (1983), 73. doi: 10.1512/iumj.1983.32.32007. Google Scholar [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. doi: 10.1007/BF00279844. Google Scholar [20] Q. Li, Constant rank theorem in complex variables,, Indiana Univ. Math. J., 58 (2009), 1235. doi: 10.1512/iumj.2009.58.3574. Google Scholar [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. doi: 10.1016/j.jfa.2008.06.008. Google Scholar [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. Google Scholar [23] J. Spruck, Geometric aspects of the theory of fully nonlinear elliptic equations, in Global Theory of Minimal Surfaces, (2005), 283. Google Scholar [24] G. Székelyhidi, Fully non-linear elliptic equations on compact Hermitian manifolds,, preprint, (). Google Scholar [25] G. Székelyhidi, V. Tosatti and B. Weinkove, Gauduchon metrics with prescribed volume form,, preprint, (). Google Scholar [26] N. S. Trudinger, Comparison principles and pointwise estimates for viscosity solutions of nonlinear elliptic equations,, Revista Mat. Iber., 4 (1988), 453. doi: 10.4171/RMI/80. Google Scholar [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. doi: 10.4171/JEMS/457. Google Scholar [28] M. Warren and Y. Yuan, Hessian estimates for the sigma-2 equation in dimension 3,, Comm. Pure Appl. Math., 62 (2009), 305. doi: 10.1002/cpa.20251. Google Scholar
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