Nonlinear oblique boundary value problems for nonlinear elliptic equations

Lieberman, Gary M.; Trudinger, Neil S.

doi:10.1090/S0002-9947-1986-0833695-6

Nonlinear oblique boundary value problems for nonlinear elliptic equations
HTML articles powered by AMS MathViewer

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

Certain properties of solutions of paraboflic equations with measurable coefficients

40

16

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

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

Solvability of the capillary problem

19

6

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

Gradient estimates for the capillary problem via the maximum principle

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

On the classical solution of Bellman’s elliptic equation

30

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