Solutions of fully nonlinear elliptic equations with patches of zero gradient: Existence, regularity and convexity of level curves

Abstract

In this paper, we first construct "viscosity" solutions (in the Crandall-Lions sense) of fully nonlinear elliptic equations of the form F(D2u, x) = g(x, u) on {|∇u| ≠ 0} In fact, viscosity solutions are surprisingly weak. Since candidates for solutions are just continuous, we only require that the "test" polynomials P (those tangent from above or below to the graph of u at a point x0) satisfy the correct inequality only if |∇P(x0)| ≠ 0. That is, we simply disregard those test polynomials for which |∇P(x0)| = 0. Nevertheless, this is enough, by an appropriate use of the Alexandroff-Bakelman technique, to prove existence, regularity and, in two dimensions, for F = Δ g = cu (c > 0) and constant boundary conditions on a convex domain, to prove that there is only one convex patch.