Constructing convex solutions via Perron’s method

Abstract

In this paper we construct convex solutions for certain elliptic boundary value problems via Perron’s method. The solutions constructed are weak solutions in the viscosity sense, and our construction follows work of Ishii (Duke Math. J., 55 (2) 369–384, 1987). The same general approach appears in work of Andrews and Feldman (J. Differential Equations, 182 (2) 298–343, 2002) in which they show existence for a weak nonlocal parabolic flow of convex curves. The time independent special case of their work leads to a one dimensional elliptic result which we extend to two dimensions. Similar results are required to extend their theory of nonlocal geometric flows to surfaces. The two dimensional case is essentially different from the one dimensional case and involves a regularity result (cf. Theorem 3.1), which has independent interest. Roughly speaking, given an arbitrary convex function (which is not smooth) supported at one point by a smooth function of prescribed Hessian (which is not convex), one must construct a third function that is both convex and smooth and appropriately approximates both of the given functions. Keywords: Viscosity solutions, Elliptic partial differential equations, Perron procedure, Convexity, Regularity, Fully nonlinear, Monge-Ampere Mathematics Subject Classification (2000:) 35J60, 53A05, 52A15, 26B05