## 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

### MSC:

 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
Full Text:

### References:

 [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)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.