A localization property of viscosity solutions to the Monge-Ampère equation and their strict convexity. (English) Zbl 0704.35045

Caffarelli, L. A.

Ann. Math. (2) 131, No. 1, 129-134 (1990).

The author considers viscosity solutions of the Monge-Ampère equation \[ (1)\quad 0<\lambda_ 1\leq \det D^ 2u\leq \lambda_ 2 \] with \(C^{1,\alpha}\) boundary data \((\alpha >1-2/n)\). Assume that \(u\geq 0\) satisfies (1) and that the convex set \(\{u=0\}\) is not a point. Then this set cannot have extremal points in the interior of the domain of definition of u.

Reviewer: G.Dziuk

MSC:

35J60	Nonlinear elliptic equations
35B50	Maximum principles in context of PDEs

Keywords:

viscosity solutions; Monge-Ampère equation; convex; extremal points