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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 2 u=f(x,u,Du) in a convex domain Ω n , where f is a prescribed positive function on Ω ¯×× n , are considered. Sufficient conditions for the existence of a unique solution to this equation, satisfying the Neumann boundary condition of the form D ν u=ϕ(x,u) are given. Authors assert that this result holds also for the standard Monge-Ampère equation det D 2 u=f(x) and the equation of prescribed Gauss curvature

detD 2 u=K(x)(1+|Du| 2 ) u+2/2 ·

Finally, this result is applied to the case when f and ϕ are independent of u.

Reviewer: V.A.Yumaguzhin
MSC:
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