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

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

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.
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 & 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,, \emph{Ann. Mat. Pura Appl.}, 58 (1962), 303.   Google Scholar

[2]

L. Barbu and C. Enache, A maximum principle for some fully nonlinear elliptic equations with applications to Weingarten surfaces,, \emph{Complex Var. Elliptic Equ.}, 58 (2013), 1725.   Google Scholar

[3]

B. Brandolini, C. Nitsch, P. Salani and C. Trombetti, Serrin type overdetermined problems : An alternative proof,, \emph{Arch. Ration. Mech. Anal.}, 190 (2008), 267.  doi: 10.1007/s00205-008-0119-3.  Google Scholar

[4]

L. A. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet Problem for Nonlinear Second-Order Elliptic Equations I. Monge-Ampère Equation,, \emph{Comm. Pure App. Math.}, 37 (1984), 369.  doi: 10.1002/cpa.3160370306.  Google Scholar

[5]

C. Enache, Maximum principles and symmetry results for a class of fully nonlinear elliptic PDEs,, \emph{NODEA - Nonlinear Differ. Equ. Appl.}, 17 (2010), 591.  doi: 10.1007/s00030-010-0070-5.  Google Scholar

[6]

C. Enache, Necessary conditions of solvability and isoperimetric estimates for some Monge-Ampère problems in the plane,, \emph{Proc. Amer. Math. Soc.}, ().   Google Scholar

[7]

E. Hopf, Elementare Bemerkung über die Lösung partieller Differentialgleichungen zweiter Ordnung vom elliptischen Typus,, \emph{Berlin Sber. Preuss. Akad. Wiss}, 19 (1927), 147.   Google Scholar

[8]

E. Hopf, A remark on linear elliptic differential equations of the second order,, \emph{Proc. Amer. Math. Soc.}, 36 (1952), 791.   Google Scholar

[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, (1000).   Google Scholar

[10]

N. M. Ivochkina, Solution of the Dirichlet problem for equations of mth order curvature, (Russian), \emph{Mat. Sb.}, 180 (1989), 867.   Google Scholar

[11]

P. L. Lions, N. S. Trudinger and J. Urbas, The Neumann problem for equations of Monge-Ampère type,, \emph{Comm. Pure Appl. Math.}, 39 (1986), 539.  doi: 10.1002/cpa.3160390405.  Google Scholar

[12]

X.-N. Ma, A necessary condition of solvability for the capillarity boundary of Monge-Ampère equations in two dimensions,, \emph{Proc. Amer. Math. Soc.}, 127 (1999), 763.  doi: 10.1090/S0002-9939-99-04750-4.  Google Scholar

[13]

X.-N. Ma, Sharp size estimates for capillary free surfaces without gravity,, \emph{Pacific J. Math.}, 192 (2000), 121.  doi: 10.2140/pjm.2000.192.121.  Google Scholar

[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,, \emph{Nonlinear Anal.}, 3 (1979), 193.  doi: 10.1016/0362-546X(79)90076-2.  Google Scholar

[15]

G. A. Philippin, A minimum principle for the problem of torsional creep,, \emph{J. Math. Anal. Appl.}, 68 (1979), 526.  doi: 10.1016/0022-247X(79)90133-1.  Google Scholar

[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,, \emph{Math. Models Meth. Appl. Sci.}, 11 (2001), 1073.  doi: 10.1142/S0218202501001240.  Google Scholar

[17]

G. A. Philippin and A. Safoui, Some applications of the maximum principle to a variety of fully nonlinear elliptic PDE's,, \emph{Z. angew. Math. Phys.}, 54 (2003), 739.  doi: 10.1007/s00033-003-3200-7.  Google Scholar

[18]

G. A. Philippin and A. Safoui, Some minimum principles for a class of elliptic boundary value problems,, \emph{Appl. Anal.}, 83 (2004), 231.  doi: 10.1080/00036810310001632754.  Google Scholar

[19]

H. Rosenberg, Hypersurfaces of constant curvature in space forms,, \emph{Bull. Sci. Math.}, 117 (1993), 211.   Google Scholar

[20]

J. Serrin, A symmetry problem in potential theory,, \emph{Arch. Ration. Mech. Anal.}, 43 (1971), 304.   Google Scholar

[21]

R. P. Sperb, Maximum Principles and Their Applications,, Academic Press, (1981).   Google Scholar

[22]

J. Urbas, Nonlinear oblique boundary value problem for Hessian equations in two dimensions,, \emph{Ann. Inst. Henri Poincare}, 12 (1995), 507.   Google Scholar

[23]

J. Urbas, A note on the contact angle boundary condition for Monge-Ampère equations,, \emph{Proc. Amer. Math. Soc.}, 128 (2000), 853.  doi: 10.1090/S0002-9939-99-05222-3.  Google Scholar

[24]

H. F. Weinberger, Remarks on the preceding paper of Serrin,, \emph{Arch. Ration. Mech. Anal.}, 43 (1971), 319.   Google Scholar

show all references

References:
[1]

A. D. Alexandrov, A characteristic property of the sphere,, \emph{Ann. Mat. Pura Appl.}, 58 (1962), 303.   Google Scholar

[2]

L. Barbu and C. Enache, A maximum principle for some fully nonlinear elliptic equations with applications to Weingarten surfaces,, \emph{Complex Var. Elliptic Equ.}, 58 (2013), 1725.   Google Scholar

[3]

B. Brandolini, C. Nitsch, P. Salani and C. Trombetti, Serrin type overdetermined problems : An alternative proof,, \emph{Arch. Ration. Mech. Anal.}, 190 (2008), 267.  doi: 10.1007/s00205-008-0119-3.  Google Scholar

[4]

L. A. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet Problem for Nonlinear Second-Order Elliptic Equations I. Monge-Ampère Equation,, \emph{Comm. Pure App. Math.}, 37 (1984), 369.  doi: 10.1002/cpa.3160370306.  Google Scholar

[5]

C. Enache, Maximum principles and symmetry results for a class of fully nonlinear elliptic PDEs,, \emph{NODEA - Nonlinear Differ. Equ. Appl.}, 17 (2010), 591.  doi: 10.1007/s00030-010-0070-5.  Google Scholar

[6]

C. Enache, Necessary conditions of solvability and isoperimetric estimates for some Monge-Ampère problems in the plane,, \emph{Proc. Amer. Math. Soc.}, ().   Google Scholar

[7]

E. Hopf, Elementare Bemerkung über die Lösung partieller Differentialgleichungen zweiter Ordnung vom elliptischen Typus,, \emph{Berlin Sber. Preuss. Akad. Wiss}, 19 (1927), 147.   Google Scholar

[8]

E. Hopf, A remark on linear elliptic differential equations of the second order,, \emph{Proc. Amer. Math. Soc.}, 36 (1952), 791.   Google Scholar

[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, (1000).   Google Scholar

[10]

N. M. Ivochkina, Solution of the Dirichlet problem for equations of mth order curvature, (Russian), \emph{Mat. Sb.}, 180 (1989), 867.   Google Scholar

[11]

P. L. Lions, N. S. Trudinger and J. Urbas, The Neumann problem for equations of Monge-Ampère type,, \emph{Comm. Pure Appl. Math.}, 39 (1986), 539.  doi: 10.1002/cpa.3160390405.  Google Scholar

[12]

X.-N. Ma, A necessary condition of solvability for the capillarity boundary of Monge-Ampère equations in two dimensions,, \emph{Proc. Amer. Math. Soc.}, 127 (1999), 763.  doi: 10.1090/S0002-9939-99-04750-4.  Google Scholar

[13]

X.-N. Ma, Sharp size estimates for capillary free surfaces without gravity,, \emph{Pacific J. Math.}, 192 (2000), 121.  doi: 10.2140/pjm.2000.192.121.  Google Scholar

[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,, \emph{Nonlinear Anal.}, 3 (1979), 193.  doi: 10.1016/0362-546X(79)90076-2.  Google Scholar

[15]

G. A. Philippin, A minimum principle for the problem of torsional creep,, \emph{J. Math. Anal. Appl.}, 68 (1979), 526.  doi: 10.1016/0022-247X(79)90133-1.  Google Scholar

[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,, \emph{Math. Models Meth. Appl. Sci.}, 11 (2001), 1073.  doi: 10.1142/S0218202501001240.  Google Scholar

[17]

G. A. Philippin and A. Safoui, Some applications of the maximum principle to a variety of fully nonlinear elliptic PDE's,, \emph{Z. angew. Math. Phys.}, 54 (2003), 739.  doi: 10.1007/s00033-003-3200-7.  Google Scholar

[18]

G. A. Philippin and A. Safoui, Some minimum principles for a class of elliptic boundary value problems,, \emph{Appl. Anal.}, 83 (2004), 231.  doi: 10.1080/00036810310001632754.  Google Scholar

[19]

H. Rosenberg, Hypersurfaces of constant curvature in space forms,, \emph{Bull. Sci. Math.}, 117 (1993), 211.   Google Scholar

[20]

J. Serrin, A symmetry problem in potential theory,, \emph{Arch. Ration. Mech. Anal.}, 43 (1971), 304.   Google Scholar

[21]

R. P. Sperb, Maximum Principles and Their Applications,, Academic Press, (1981).   Google Scholar

[22]

J. Urbas, Nonlinear oblique boundary value problem for Hessian equations in two dimensions,, \emph{Ann. Inst. Henri Poincare}, 12 (1995), 507.   Google Scholar

[23]

J. Urbas, A note on the contact angle boundary condition for Monge-Ampère equations,, \emph{Proc. Amer. Math. Soc.}, 128 (2000), 853.  doi: 10.1090/S0002-9939-99-05222-3.  Google Scholar

[24]

H. F. Weinberger, Remarks on the preceding paper of Serrin,, \emph{Arch. Ration. Mech. Anal.}, 43 (1971), 319.   Google Scholar

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