THE OBSTACLE PROBLEM FOR A DEGENERATE FULLY NONLINEAR OPERATORS

JOAO DA SILVA; HERNÁN VIVAS

Río Cuarto

Congreso; Congreso VII MACI 2019; 2019

ASAMACI

We study regularity estimates for viscosity solutions of an obstacle-type problem driven by a fully nonlinear elliptic operator with a degenerate character: \begin{equation*} \left\{ \begin{array}{rcll} \min\left\{f-|D u|^\gamma F(D^2u),u-\phi\right\} & = & 0 & \textrm{ in } \Omega \\ u & = & g & \textrm{ on } \partial \Omega,\end{array}\right.\end{equation*}Here $\Omega \subset \R^n$ is a bounded and regular domain, $g$ is continuous on $\partial\Omega$, $\phi$ is a given (regular) obstacle, $\gamma >0$, $f$ is a continuous and bounded function in $\Omega$ and $F$ is a fully nonlinear uniformly elliptic operator. We prove that solutions enjoy the same regularity as the obstacle along free boundary points, even if this regularity is higher than the one constrained by the operator. Furthermore, for the homogeneous case we obtain a non-degeneracy property that implies that the free boundary has zero Lebesgue measure.