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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
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