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 the obstacle problem for fully nonlinear elliptic operators with an anisotropic degeneracy on the gradient:\[ \left\{ \begin{array}{rcll} \min\left\{f-|Du|^\gamma F(D^2u),u-\phi\right\} & = & 0 & \textrm{ in } \Omega \\ u & = & g & \textrm{ on } \partial \Omega \end{array}\right.\]for some degeneracy parameter $\gamma\geq 0$, uniformly elliptic operator $F$, bounded source term $f$, and suitably smooth obstacle $\phi$ and boundary datum $g$. We obtain existence/uniqueness of solutions and prove sharp regularity estimates at the free boundary points, namely $\partial\{u>\phi\} \cap \Omega$. In particular, for the homogeneous case ($f\equiv0$) we get that solutions are $C^{1,1}$ at free boundary points, in the sense that they detach from the obstacle in a quadratic fashion, thus beating the optimal regularity allowed for such degenerate operators. We also prove several non-degeneracy properties of solutions and partial results regarding the free boundary.These are the first results for obstacle problems driven by degenerate type operators in non-divergence form and they are a novelty even for the simpler prototype given by an operator of the form $\mathcal{G}[u] = |Du|^\gamma\Delta u$, with $\gamma >0$ and $f \equiv 1$.