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Revista Matemática Iberoamericana

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Published online: 2021-01-21

The obstacle problem for a class of degenerate fully nonlinear operators

João Vítor da Silva^[1] and Hernán Vivas^[2]

(1) Universidade Estadual de Campinas, Brazil
(2) Universidad de Buenos Aires and Centro Marplatense de Investigaciones Matemáticas, Mar del Plata, Argentina

We study the obstacle problem for fully nonlinear elliptic operators with an anisotropic degeneracy on the gradient: \[ \left\{\begin{array}{rll} \min\left\{f-|Du|^\gamma F(D^2u),u-\phi\right\} &= 0 & \textrm{ in } \Omega,\\ u & = g & \textrm{ on } \partial \Omega, \end{array}\right. \] 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 $f \equiv 1$.

Keywords: Free boundary problems, degenerate elliptic equations

da Silva João Vítor, Vivas Hernán: The obstacle problem for a class of degenerate fully nonlinear operators. Rev. Mat. Iberoam. 37 (2021), 1991-2020. doi: 10.4171/rmi/1256