# American Institute of Mathematical Sciences

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 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.

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##### 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.
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