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

In this paper, we first construct "viscosity" solutions (in the Crandall-Lions sense) of fully nonlinear elliptic equations of the form F(D(2)u, x) = g(x, u) on {\delu\ not equal 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 x(0)) satisfy the correct inequality only if \delP(x(0))\ not equal 0. That is, we simply disregard those test polynomials for which \delP(x(0))\ = 0. Nevertheless, this is enough, by an appropriate use of the Alexandroff-Bakelman technique, to prove existence, regularity and, in two dimensions, for F = Delta, g = cu (c > 0) and constant boundary conditions on a convex domain, to prove that there is only one convex patch.