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Solutions of fully nonlinear elliptic equations with patches of zero gradient: Existence, regularity and convexity of level curves
In this paper, we first construct âviscosityâ 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 …