The Neumann problem for equations of Monge-Ampère type. (English) Zbl 0604.35027

The paper is concerned with the existence of classical solutions of semilinear Neumann problems for equations of Monge-Ampère type: \[ \det D^ 2u=f(x,u,Du)\quad in\quad G\subset {\mathbb{R}}^ n;\quad D_{\nu}u=g(x,u)\quad on\quad \partial G. \] For uniformly convex domains G with smooth boundary the main theorem gives conditions on f and g which guarantee the existence and uniqueness of a convex solution \(u\in C^{3,\alpha}(\bar G)\) for all \(\alpha <1\). The proof employs the method of continuity which requires a priori estimates of u in \(C^{2,\alpha}(\bar G)\) for some \(\alpha >0\). The estimation of u and Du is done for general boundary conditions which include the Dirichlet problem as a special case. The techniques used to estimate \(D^ 2 u\) for Neumann problems differ considerably from those known for Dirichlet problems. The last section of the paper indicates generalizations, variants and applications of the main theorem.
Reviewer: J.Weisel


35J65 Nonlinear boundary value problems for linear elliptic equations
35B45 A priori estimates in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
35J25 Boundary value problems for second-order elliptic equations
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[1] Bakel’man, Dokl. Akad. Nauk., SSSR. 126 pp 923–
[2] The Dirichlet problem for the elliptic n-dimensional Monge-Ampère equations and related problems in the theory of quasilinear equations, Proceedings of Seminar on Monge-Ampère Equations and Related Topics, Firenze, 1980, Instituto Nazionale di Alta Matematica, Roma, 1982, pp. 1–78.
[3] Caffarelli, Comm. Pure Appl. Math. 37 pp 369– (1984)
[4] Cheng, Comm. Pure Appl. Math. 30 pp 41– (1977)
[5] and , Elliptic partial differential equations of second order, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, Second Edition, 1983. · Zbl 0361.35003
[6] Ivochkina, Zap. Naucn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 96 pp 69– (1980)
[7] J. Soviet Math. 21 pp 689– (1983)
[8] Ivochkina, Zap. Naucn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 131 pp 72– (1983)
[9] Krylov, Izv. Akad. Nauk. SSSR 47 pp 75– (1983)
[10] Lieberman, Trans. Amer. Math. Soc. 296 (1986)
[11] Lions, Manuscripta Math. 41 pp 1– (1983)
[12] Lions, Arch. Rational Mech. Anal. 89 pp 93– (1985)
[13] Lions, Duke Math. J. 52 pp 793– (1985)
[14] Lions, J. Analyse Math.
[15] Lions, Math. Zeit 191 pp 1– (1986)
[16] Pogorelov, Dokl. Akad. Nauk. SSSR 201 pp 790– (1971)
[17] Soviet Math. Dokl. 12 pp 1727– (1971)
[18] Trudinger, Proc. Centre for Math. Anal. Aus. Nat. Univ. 8 pp 65– (1984)
[19] Trudinger, Bull. Austral. Math. Soc. 28 pp 217– (1983)
[20] Urbas, J. Differential Geometry 20 pp 311– (1984)
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