Maximum and minimum principles for a class of Monge-Ampère equations in the plane, with applications to surfaces of constant Gauss curvature

Previous Article
CPAA Home
This Issue
Next Article
Global existence of strong solutions to incompressible MHD

May 2014, 13(3): 1347-1359. doi: 10.3934/cpaa.2014.13.1347

Maximum and minimum principles for a class of Monge-Ampère equations in the plane, with applications to surfaces of constant Gauss curvature

Cristian Enache ^1,

1.	Research group of the project PN-II-ID-PCE-2012-4-0021, "Simion Stoilow" Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, 014700 Bucharest, Romania

Received September 2013 Revised November 2013 Published December 2013

Abstract
Related Papers

In this article we investigate a general class of Monge-Ampère equations in the plane, including the constant Gauss curvature equation. Our first aim is to prove some maximum and minimum principles for suitable $P$-functions, in the sense of L.E. Payne. Then, these new principles are employed to solve a general class of overdetermined Monge-Ampère problems and to investigate two boundary value problems for the constant Gauss curvature equation. More precisely, when the constant Gauss curvature equation is subject to the homogeneous Dirichlet boundary condition, we prove several isoperimetric inequalities, while when it is subject to the contact angle boundary condition, some necessary conditions of solvability, involving the curvature of the boundary of the underlying domain and the given contact angle, are derived.

Keywords: isoperimetric estimates., Monge-Ampère problems, overdetermined problems, maximum principles.

Mathematics Subject Classification: Primary: 35B50; Secondary: 35J25, 35P15.

Citation: Cristian Enache. Maximum and minimum principles for a class of Monge-Ampère equations in the plane, with applications to surfaces of constant Gauss curvature. Communications on Pure and Applied Analysis, 2014, 13 (3) : 1347-1359. doi: 10.3934/cpaa.2014.13.1347

References:

[1]	A. D. Alexandrov, A characteristic property of the sphere, Ann. Mat. Pura Appl., 58 (1962), 303-354.
[2]	L. Barbu and C. Enache, A maximum principle for some fully nonlinear elliptic equations with applications to Weingarten surfaces, Complex Var. Elliptic Equ., 58 (2013), 1725-1736.
[3]	B. Brandolini, C. Nitsch, P. Salani and C. Trombetti, Serrin type overdetermined problems : An alternative proof, Arch. Ration. Mech. Anal., 190 (2008), 267-280. doi: 10.1007/s00205-008-0119-3.
[4]	L. A. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet Problem for Nonlinear Second-Order Elliptic Equations I. Monge-Ampère Equation, Comm. Pure App. Math., 37 (1984), 369-402. doi: 10.1002/cpa.3160370306.
[5]	C. Enache, Maximum principles and symmetry results for a class of fully nonlinear elliptic PDEs, NODEA - Nonlinear Differ. Equ. Appl., 17 (2010), 591-600. doi: 10.1007/s00030-010-0070-5.
[6]	C. Enache, Necessary conditions of solvability and isoperimetric estimates for some Monge-Ampère problems in the plane,, \emph{Proc. Amer. Math. Soc.}, ().
[7]	E. Hopf, Elementare Bemerkung über die Lösung partieller Differentialgleichungen zweiter Ordnung vom elliptischen Typus, Berlin Sber. Preuss. Akad. Wiss, 19 (1927), 147-152.
[8]	E. Hopf, A remark on linear elliptic differential equations of the second order, Proc. Amer. Math. Soc., 36 (1952), 791-793.
[9]	H. Hopf, Differential geometry in the large: seminar lectures New York University 1946 and Stanford University 1956, vol. 1000 of Lecture Notes in Mathematics, Springer Verlag, 1989.
[10]	N. M. Ivochkina, Solution of the Dirichlet problem for equations of mth order curvature, (Russian) Mat. Sb., 180 (1989), 867-887; translation in Math. USSR-Sb., 67 (1990), 317-339.
[11]	P. L. Lions, N. S. Trudinger and J. Urbas, The Neumann problem for equations of Monge-Ampère type, Comm. Pure Appl. Math., 39 (1986), 539-563. doi: 10.1002/cpa.3160390405.
[12]	X.-N. Ma, A necessary condition of solvability for the capillarity boundary of Monge-Ampère equations in two dimensions, Proc. Amer. Math. Soc., 127 (1999), 763-769. doi: 10.1090/S0002-9939-99-04750-4.
[13]	X.-N. Ma, Sharp size estimates for capillary free surfaces without gravity, Pacific J. Math., 192 (2000), 121-134. doi: 10.2140/pjm.2000.192.121.
[14]	L. E. Payne and G. A. Philippin, Some maximum principles for nonlinear elliptic equations in divergence form with applications to capillary surfaces and to surfaces of constant mean curvature, Nonlinear Anal., 3 (1979), 193-211. doi: 10.1016/0362-546X(79)90076-2.
[15]	G. A. Philippin, A minimum principle for the problem of torsional creep, J. Math. Anal. Appl., 68 (1979), 526-535. doi: 10.1016/0022-247X(79)90133-1.
[16]	G. A. Philippin and A. Safoui, Some maximum principles and symmetry results for a class of boundary value problems involving the Monge-Ampère equation, Math. Models Meth. Appl. Sci., 11 (2001), 1073-1080. doi: 10.1142/S0218202501001240.
[17]	G. A. Philippin and A. Safoui, Some applications of the maximum principle to a variety of fully nonlinear elliptic PDE's, Z. angew. Math. Phys., 54 (2003), 739-755. doi: 10.1007/s00033-003-3200-7.
[18]	G. A. Philippin and A. Safoui, Some minimum principles for a class of elliptic boundary value problems, Appl. Anal., 83 (2004), 231-241. doi: 10.1080/00036810310001632754.
[19]	H. Rosenberg, Hypersurfaces of constant curvature in space forms, Bull. Sci. Math., 117 (1993), 211-239.
[20]	J. Serrin, A symmetry problem in potential theory, Arch. Ration. Mech. Anal., 43 (1971), 304-318.
[21]	R. P. Sperb, Maximum Principles and Their Applications, Academic Press, (1981).
[22]	J. Urbas, Nonlinear oblique boundary value problem for Hessian equations in two dimensions, Ann. Inst. Henri Poincare, 12 (1995), 507-575.
[23]	J. Urbas, A note on the contact angle boundary condition for Monge-Ampère equations, Proc. Amer. Math. Soc., 128 (2000), 853-855. doi: 10.1090/S0002-9939-99-05222-3.
[24]	H. F. Weinberger, Remarks on the preceding paper of Serrin, Arch. Ration. Mech. Anal., 43 (1971), 319-320.

show all references

References:

[1]	A. D. Alexandrov, A characteristic property of the sphere, Ann. Mat. Pura Appl., 58 (1962), 303-354.
[2]	L. Barbu and C. Enache, A maximum principle for some fully nonlinear elliptic equations with applications to Weingarten surfaces, Complex Var. Elliptic Equ., 58 (2013), 1725-1736.
[3]	B. Brandolini, C. Nitsch, P. Salani and C. Trombetti, Serrin type overdetermined problems : An alternative proof, Arch. Ration. Mech. Anal., 190 (2008), 267-280. doi: 10.1007/s00205-008-0119-3.
[4]	L. A. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet Problem for Nonlinear Second-Order Elliptic Equations I. Monge-Ampère Equation, Comm. Pure App. Math., 37 (1984), 369-402. doi: 10.1002/cpa.3160370306.
[5]	C. Enache, Maximum principles and symmetry results for a class of fully nonlinear elliptic PDEs, NODEA - Nonlinear Differ. Equ. Appl., 17 (2010), 591-600. doi: 10.1007/s00030-010-0070-5.
[6]	C. Enache, Necessary conditions of solvability and isoperimetric estimates for some Monge-Ampère problems in the plane,, \emph{Proc. Amer. Math. Soc.}, ().
[7]	E. Hopf, Elementare Bemerkung über die Lösung partieller Differentialgleichungen zweiter Ordnung vom elliptischen Typus, Berlin Sber. Preuss. Akad. Wiss, 19 (1927), 147-152.
[8]	E. Hopf, A remark on linear elliptic differential equations of the second order, Proc. Amer. Math. Soc., 36 (1952), 791-793.
[9]	H. Hopf, Differential geometry in the large: seminar lectures New York University 1946 and Stanford University 1956, vol. 1000 of Lecture Notes in Mathematics, Springer Verlag, 1989.
[10]	N. M. Ivochkina, Solution of the Dirichlet problem for equations of mth order curvature, (Russian) Mat. Sb., 180 (1989), 867-887; translation in Math. USSR-Sb., 67 (1990), 317-339.
[11]	P. L. Lions, N. S. Trudinger and J. Urbas, The Neumann problem for equations of Monge-Ampère type, Comm. Pure Appl. Math., 39 (1986), 539-563. doi: 10.1002/cpa.3160390405.
[12]	X.-N. Ma, A necessary condition of solvability for the capillarity boundary of Monge-Ampère equations in two dimensions, Proc. Amer. Math. Soc., 127 (1999), 763-769. doi: 10.1090/S0002-9939-99-04750-4.
[13]	X.-N. Ma, Sharp size estimates for capillary free surfaces without gravity, Pacific J. Math., 192 (2000), 121-134. doi: 10.2140/pjm.2000.192.121.
[14]	L. E. Payne and G. A. Philippin, Some maximum principles for nonlinear elliptic equations in divergence form with applications to capillary surfaces and to surfaces of constant mean curvature, Nonlinear Anal., 3 (1979), 193-211. doi: 10.1016/0362-546X(79)90076-2.
[15]	G. A. Philippin, A minimum principle for the problem of torsional creep, J. Math. Anal. Appl., 68 (1979), 526-535. doi: 10.1016/0022-247X(79)90133-1.
[16]	G. A. Philippin and A. Safoui, Some maximum principles and symmetry results for a class of boundary value problems involving the Monge-Ampère equation, Math. Models Meth. Appl. Sci., 11 (2001), 1073-1080. doi: 10.1142/S0218202501001240.
[17]	G. A. Philippin and A. Safoui, Some applications of the maximum principle to a variety of fully nonlinear elliptic PDE's, Z. angew. Math. Phys., 54 (2003), 739-755. doi: 10.1007/s00033-003-3200-7.
[18]	G. A. Philippin and A. Safoui, Some minimum principles for a class of elliptic boundary value problems, Appl. Anal., 83 (2004), 231-241. doi: 10.1080/00036810310001632754.
[19]	H. Rosenberg, Hypersurfaces of constant curvature in space forms, Bull. Sci. Math., 117 (1993), 211-239.
[20]	J. Serrin, A symmetry problem in potential theory, Arch. Ration. Mech. Anal., 43 (1971), 304-318.
[21]	R. P. Sperb, Maximum Principles and Their Applications, Academic Press, (1981).
[22]	J. Urbas, Nonlinear oblique boundary value problem for Hessian equations in two dimensions, Ann. Inst. Henri Poincare, 12 (1995), 507-575.
[23]	J. Urbas, A note on the contact angle boundary condition for Monge-Ampère equations, Proc. Amer. Math. Soc., 128 (2000), 853-855. doi: 10.1090/S0002-9939-99-05222-3.
[24]	H. F. Weinberger, Remarks on the preceding paper of Serrin, Arch. Ration. Mech. Anal., 43 (1971), 319-320.

[1]	Jiakun Liu, Neil S. Trudinger. On Pogorelov estimates for Monge-Ampère type equations. Discrete and Continuous Dynamical Systems, 2010, 28 (3) : 1121-1135. doi: 10.3934/dcds.2010.28.1121
[2]	Qi-Rui Li, Xu-Jia Wang. Regularity of the homogeneous Monge-Ampère equation. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 6069-6084. doi: 10.3934/dcds.2015.35.6069
[3]	Diego Maldonado. On interior $C^2$-estimates for the Monge-Ampère equation. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 1427-1440. doi: 10.3934/dcds.2018058
[4]	Barbara Brandolini, Carlo Nitsch, Cristina Trombetti. Shape optimization for Monge-Ampère equations via domain derivative. Discrete and Continuous Dynamical Systems - S, 2011, 4 (4) : 825-831. doi: 10.3934/dcdss.2011.4.825
[5]	Yahui Niu. Monotonicity of solutions for a class of nonlocal Monge-Ampère problem. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5269-5283. doi: 10.3934/cpaa.2020237
[6]	Luca Codenotti, Marta Lewicka. Visualization of the convex integration solutions to the Monge-Ampère equation. Evolution Equations and Control Theory, 2019, 8 (2) : 273-300. doi: 10.3934/eect.2019015
[7]	Limei Dai, Hongyu Li. Entire subsolutions of Monge-Ampère type equations. Communications on Pure and Applied Analysis, 2020, 19 (1) : 19-30. doi: 10.3934/cpaa.2020002
[8]	Bo Guan, Qun Li. A Monge-Ampère type fully nonlinear equation on Hermitian manifolds. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 1991-1999. doi: 10.3934/dcdsb.2012.17.1991
[9]	Fan Cui, Huaiyu Jian. Symmetry of solutions to a class of Monge-Ampère equations. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1247-1259. doi: 10.3934/cpaa.2019060
[10]	Juhua Shi, Feida Jiang. The degenerate Monge-Ampère equations with the Neumann condition. Communications on Pure and Applied Analysis, 2021, 20 (2) : 915-931. doi: 10.3934/cpaa.2020297
[11]	Alessio Figalli, Young-Heon Kim. Partial regularity of Brenier solutions of the Monge-Ampère equation. Discrete and Continuous Dynamical Systems, 2010, 28 (2) : 559-565. doi: 10.3934/dcds.2010.28.559
[12]	Limei Dai. Multi-valued solutions to a class of parabolic Monge-Ampère equations. Communications on Pure and Applied Analysis, 2014, 13 (3) : 1061-1074. doi: 10.3934/cpaa.2014.13.1061
[13]	Adam M. Oberman. Wide stencil finite difference schemes for the elliptic Monge-Ampère equation and functions of the eigenvalues of the Hessian. Discrete and Continuous Dynamical Systems - B, 2008, 10 (1) : 221-238. doi: 10.3934/dcdsb.2008.10.221
[14]	Zhijun Zhang. Optimal global asymptotic behavior of the solution to a singular monge-ampère equation. Communications on Pure and Applied Analysis, 2020, 19 (2) : 1129-1145. doi: 10.3934/cpaa.2020053
[15]	Shuyu Gong, Ziwei Zhou, Jiguang Bao. Existence and uniqueness of viscosity solutions to the exterior problem of a parabolic Monge-Ampère equation. Communications on Pure and Applied Analysis, 2020, 19 (10) : 4921-4936. doi: 10.3934/cpaa.2020218
[16]	Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure and Applied Analysis, 2021, 20 (1) : 301-317. doi: 10.3934/cpaa.2020267
[17]	Jingang Xiong, Jiguang Bao. The obstacle problem for Monge-Ampère type equations in non-convex domains. Communications on Pure and Applied Analysis, 2011, 10 (1) : 59-68. doi: 10.3934/cpaa.2011.10.59
[18]	Shouchuan Hu, Haiyan Wang. Convex solutions of boundary value problem arising from Monge-Ampère equations. Discrete and Continuous Dynamical Systems, 2006, 16 (3) : 705-720. doi: 10.3934/dcds.2006.16.705
[19]	Ziwei Zhou, Jiguang Bao, Bo Wang. A Liouville theorem of parabolic Monge-AmpÈre equations in half-space. Discrete and Continuous Dynamical Systems, 2021, 41 (4) : 1561-1578. doi: 10.3934/dcds.2020331
[20]	Nam Q. Le. Optimal boundary regularity for some singular Monge-Ampère equations on bounded convex domains. Discrete and Continuous Dynamical Systems, 2022, 42 (5) : 2199-2214. doi: 10.3934/dcds.2021188

2020 Impact Factor: 1.916

Tools

Metrics

Other articles
by authors

on AIMS
Cristian Enache
on Google Scholar
Cristian Enache

[Back to Top]