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 and Continuous Dynamical Systems, 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), 265-288. 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 $\mathbb{R}^3$, 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.

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), 265-288. 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 $\mathbb{R}^3$, 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.

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