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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 ${D}^{2}u=f\left(x,u,Du\right)$ in a convex domain ${\Omega }\subset {ℝ}^{n}$, where f is a prescribed positive function on $\overline{{\Omega }}×ℝ×{ℝ}^{n}$, are considered. Sufficient conditions for the existence of a unique solution to this equation, satisfying the Neumann boundary condition of the form ${D}_{\nu }u=\phi \left(x,u\right)$ are given. Authors assert that this result holds also for the standard Monge-Ampère equation det ${D}^{2}u=f\left(x\right)$ and the equation of prescribed Gauss curvature

$det{D}^{2}u={K\left(x\right)\left(1+|Du|}^{2}{\right)}^{u+2/2}·$

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

Reviewer: V.A.Yumaguzhin
##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 35G30 Boundary value problems for nonlinear higher-order PDE 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 53A10 Minimal surfaces, surfaces with prescribed mean curvature