Nonlinear oblique boundary value problems for nonlinear elliptic equations
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- by Gary M. Lieberman and Neil S. Trudinger PDF
- Trans. Amer. Math. Soc. 295 (1986), 509-546 Request permission
Abstract:
We consider the nonlinear oblique derivative boundary value problem for quasilinear and fully nonlinear uniformly elliptic partial differential equations of second order. The elliptic operators satisfy natural structure conditions as introduced by Trudinger in the study of the Dirichlet problem while for the boundary operators we formulate general structure conditions which embrace previously considered special cases such as the capillarity condition. The resultant existence theorems include previous work such as that of Lieberman on quasilinear equations and Lions and Trudinger on Neumann boundary conditions.References
- S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math. 12 (1959), 623–727. MR 125307, DOI 10.1002/cpa.3160120405
- L. Caffarelli, L. Nirenberg, and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations. I. Monge-Ampère equation, Comm. Pure Appl. Math. 37 (1984), no. 3, 369–402. MR 739925, DOI 10.1002/cpa.3160370306
- Lawrence C. Evans, Classical solutions of fully nonlinear, convex, second-order elliptic equations, Comm. Pure Appl. Math. 35 (1982), no. 3, 333–363. MR 649348, DOI 10.1002/cpa.3160350303
- Lawrence C. Evans and Avner Friedman, Optimal stochastic switching and the Dirichlet problem for the Bellman equation, Trans. Amer. Math. Soc. 253 (1979), 365–389. MR 536953, DOI 10.1090/S0002-9947-1979-0536953-4
- Lawrence C. Evans and Pierre-Louis Lions, Résolution des équations de Hamilton-Jacobi-Bellman pour des opérateurs uniformément elliptiques, C. R. Acad. Sci. Paris Sér. A-B 290 (1980), no. 22, A1049–A1052 (French, with English summary). MR 582494
- Renato Fiorenza, Sui problemi di derivata obliqua per le equazioni ellittiche quasi lineari, Ricerche Mat. 15 (1966), 74–108 (Italian). MR 216160
- Claus Gerhardt, Boundary value problems for surfaces of prescribed mean curvature, J. Math. Pures Appl. (9) 58 (1979), no. 1, 75–109. MR 533236
- David Gilbarg and Lars Hörmander, Intermediate Schauder estimates, Arch. Rational Mech. Anal. 74 (1980), no. 4, 297–318. MR 588031, DOI 10.1007/BF00249677
- David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190, DOI 10.1007/978-3-642-61798-0
- L. I. Kamynin and B. N. Himčenko, A maximum principle and Lipschitz boundary estimates for the solution of a second order elliptic-parabolic equation, Sibirsk. Mat. Ž. 15 (1974), 343–367, 461 (Russian). MR 0437947
- N. V. Krylov, Boundedly inhomogeneous elliptic and parabolic equations, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 3, 487–523, 670 (Russian). MR 661144
- N. V. Krylov, Boundedly inhomogeneous elliptic and parabolic equations in a domain, Izv. Akad. Nauk SSSR Ser. Mat. 47 (1983), no. 1, 75–108 (Russian). MR 688919 N. V. Krylov and M. V. Safonov, Certain properties of solutions of paraboflic equations with measurable coefficients, Izv. Akad. Nauk SSSR 40 (1980), 161-175; English transl., Math. USSR Izv. 16 (1981), 151-164.
- Olga A. Ladyzhenskaya and Nina N. Ural’tseva, Linear and quasilinear elliptic equations, Academic Press, New York-London, 1968. Translated from the Russian by Scripta Technica, Inc; Translation editor: Leon Ehrenpreis. MR 0244627
- Gary M. Lieberman, The quasilinear Dirichlet problem with decreased regularity at the boundary, Comm. Partial Differential Equations 6 (1981), no. 4, 437–497. MR 612553, DOI 10.1080/03605308108820184
- Gary M. Lieberman, Solvability of quasilinear elliptic equations with nonlinear boundary conditions, Trans. Amer. Math. Soc. 273 (1982), no. 2, 753–765. MR 667172, DOI 10.1090/S0002-9947-1982-0667172-9
- Gary M. Lieberman, The conormal derivative problem for elliptic equations of variational type, J. Differential Equations 49 (1983), no. 2, 218–257. MR 708644, DOI 10.1016/0022-0396(83)90013-X
- Gary M. Lieberman, The nonlinear oblique derivative problem for quasilinear elliptic equations, Nonlinear Anal. 8 (1984), no. 1, 49–65. MR 732415, DOI 10.1016/0362-546X(84)90027-0
- Gary M. Lieberman, Solvability of quasilinear elliptic equations with nonlinear boundary conditions. II, J. Funct. Anal. 56 (1984), no. 2, 210–219. MR 738579, DOI 10.1016/0022-1236(84)90087-9
- Gary M. Lieberman, Regularized distance and its applications, Pacific J. Math. 117 (1985), no. 2, 329–352. MR 779924
- Gary M. Lieberman, Intermediate Schauder estimates for oblique derivative problems, Arch. Rational Mech. Anal. 93 (1986), no. 2, 129–134. MR 823115, DOI 10.1007/BF00279956
- Gary M. Lieberman, The Dirichlet problem for quasilinear elliptic equations with continuously differentiable boundary data, Comm. Partial Differential Equations 11 (1986), no. 2, 167–229. MR 818099, DOI 10.1080/03605308608820422
- Gary M. Lieberman, Regularity of solutions of nonlinear elliptic boundary value problems, J. Reine Angew. Math. 369 (1986), 1–13. MR 850625, DOI 10.1515/crll.1986.369.1
- P.-L. Lions, Résolution de problèmes elliptiques quasilinéaires, Arch. Rational Mech. Anal. 74 (1980), no. 4, 335–353 (French). MR 588033, DOI 10.1007/BF00249679
- P.-L. Lions, Résolution analytique des problèmes de Bellman-Dirichlet, Acta Math. 146 (1981), no. 3-4, 151–166 (French). MR 611381, DOI 10.1007/BF02392461 —, Hamilton-Jacobi-Bellman equations and the optimal control of stochastic systems, Proc. Internat. Congress Math. Warsaw, 1983.
- P.-L. Lions and N. S. Trudinger, Linear oblique derivative problems for the uniformly elliptic Hamilton-Jacobi-Bellman equation, Math. Z. 191 (1986), no. 1, 1–15. MR 812598, DOI 10.1007/BF01163605
- Leon Simon and Joel Spruck, Existence and regularity of a capillary surface with prescribed contact angle, Arch. Rational Mech. Anal. 61 (1976), no. 1, 19–34. MR 487724, DOI 10.1007/BF00251860
- Neil S. Trudinger, Local estimates for subsolutions and supersolutions of general second order elliptic quasilinear equations, Invent. Math. 61 (1980), no. 1, 67–79. MR 587334, DOI 10.1007/BF01389895
- Neil S. Trudinger, Fully nonlinear, uniformly elliptic equations under natural structure conditions, Trans. Amer. Math. Soc. 278 (1983), no. 2, 751–769. MR 701522, DOI 10.1090/S0002-9947-1983-0701522-0
- Neil S. Trudinger, Regularity of solutions of fully nonlinear elliptic equations, Boll. Un. Mat. Ital. A (6) 3 (1984), no. 3, 421–430. MR 769173 N. N. Ural’tseva, Solvability of the capillary problem, Vestnik Leningrad. Univ. 19 (1973), 54-64; English transl., Vestnik Leningrad Univ. Math. 6 (1979), 363-375.
- Wolf von Wahl, Über quasilineare elliptische Differentialgleichungen in der Ebene, Manuscripta Math. 8 (1973), 59–67 (German, with English summary). MR 330747, DOI 10.1007/BF01317577 N. Korevaar and G. M. Lieberman, Gradient estimates for the capillary problem via the maximum principle (to appear).
- Gary M. Lieberman, Gradient bounds for solutions of nonuniformly elliptic oblique derivative problems, Nonlinear Anal. 11 (1987), no. 1, 49–61. MR 872039, DOI 10.1016/0362-546X(87)90025-3
- Gary M. Lieberman, Two-dimensional nonlinear boundary value problems for elliptic equations, Trans. Amer. Math. Soc. 300 (1987), no. 1, 287–295. MR 871676, DOI 10.1090/S0002-9947-1987-0871676-8
- J. D. Madjarova, On the Neumann problem for a fully nonlinear convex elliptic equation, C. R. Acad. Bulgare Sci. 38 (1985), no. 2, 183–186. MR 789129
- N. S. Nadirashvili, On a problem with an oblique derivative, Mat. Sb. (N.S.) 127(169) (1985), no. 3, 398–416 (Russian). MR 798384 M. V. Safonov, On the classical solution of Bellman’s elliptic equation, Soviet Math. Dokl. 30 (1984), 482-485.
- Neil S. Trudinger, Boundary value problems for fully nonlinear elliptic equations, Miniconference on nonlinear analysis (Canberra, 1984) Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 8, Austral. Nat. Univ., Canberra, 1984, pp. 65–83. MR 799213
- Neil S. Trudinger, On an interpolation inequality and its applications to nonlinear elliptic equations, Proc. Amer. Math. Soc. 95 (1985), no. 1, 73–78. MR 796449, DOI 10.1090/S0002-9939-1985-0796449-X —, Lectures on nonlinear second order elliptic equations, Nankai Institute of Mathematics, Tiajin, China, 1985.
Additional Information
- © Copyright 1986 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 295 (1986), 509-546
- MSC: Primary 35J65
- DOI: https://doi.org/10.1090/S0002-9947-1986-0833695-6
- MathSciNet review: 833695