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 \begin{equation*}F(D^{2} u,x) = g(x,u)\ \text { on }\ \{|\nabla u| \ne 0\}\end{equation*} 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 $|\nabla P (x_{0})| \ne 0$. That is, we simply disregard those test polynomials for which $|\nabla P (x_{0})| = 0$. 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.