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The Neumann problem for equations of Monge-Ampère type. (English) Zbl 0617.35050
Geometry and partial differential equations, Miniconf. Canberra/Aust. 1985, Proc. Cent. Math. Anal. Aust. Natl. Univ. 10, 135-140 (1986).
[For the entire collection see Zbl 0583.00013.] Equations of Monge-Ampère type det $D\sp 2 u=f(x,u,Du)$ in a convex domain $\Omega \subset {\bbfR}\sp n$, where f is a prescribed positive function on ${\bar \Omega}\times {\bbfR}\times {\bbfR}\sp n$, are considered. Sufficient conditions for the existence of a unique solution to this equation, satisfying the Neumann boundary condition of the form $D\sb{\nu}u=\phi (x,u)$ are given. Authors assert that this result holds also for the standard Monge-Ampère equation det $D\sp 2 u=f(x)$ and the equation of prescribed Gauss curvature $$ \det D\sp 2 u=K(x)(1+\vert Du\vert\sp 2)\sp{u+2/2}. $$ Finally, this result is applied to the case when f and $\phi$ are independent of u.
Reviewer: V.A.Yumaguzhin

35J65Nonlinear boundary value problems for linear elliptic equations
35G30Boundary value problems for nonlinear higher-order PDE
35A05General existence and uniqueness theorems (PDE) (MSC2000)
53A10Minimal surfaces, surfaces with prescribed mean curvature