# The obstacle problem for a class of degenerate fully nonlinear operators

### João Vítor da Silva

Universidade Estadual de Campinas, Brazil### Hernán Vivas

Universidad de Buenos Aires and Centro Marplatense de Investigaciones Matemáticas, Mar del Plata, Argentina

## Abstract

We study the obstacle problem for fully nonlinear elliptic operators with an anisotropic degeneracy on the gradient:

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 $f \equiv 1$.

## Cite this article

João Vítor da Silva, Hernán Vivas, The obstacle problem for a class of degenerate fully nonlinear operators. Rev. Mat. Iberoam. 37 (2021), no. 5, pp. 1991–2020

DOI 10.4171/RMI/1256