A limiting free boundary problem with gradient constraint and Tug-of-War games

Abstract

In this manuscript we deal with regularity issues and the asymptotic behaviour (as $p \to \infty$) of solutions for elliptic free boundary problems of $p-$Laplacian type ($2 \leq p< \infty$): \begin{equation*} -\Delta_p u(x) + \lambda_0(x)\chi_{\{u>0\}}(x) = 0 \quad \mbox{in} \quad \Omega \subset \mathbb{R}^N, \end{equation*} with a prescribed Dirichlet boundary data, where $\lambda_0>0$ is a bounded function and $\Omega$ is a regular domain. First, we prove the convergence as $p\to \infty$ of any family of solutions $(u_p)_{p\geq 2}$, as well as we obtain the corresponding limit operator (in non-divergence form) ruling the limit equation, $$ \left\{ \begin{array}{rcrcl} \max\left\{-\Delta_{\infty} u_{\infty}, \,\, -|\nabla u_{\infty}| + \chi_{\{u_{\infty}>0\}}\right\} & = & 0 & \text{in} & \Omega \cap \{u_{\infty} \geq 0\} \\ u_{\infty} & = & g & \text{on} & \partial \Omega. \end{array} \right. $$ Next, we obtain uniqueness for solutions to this limit problem together with a number of weak geometric and measure theoretical properties as non-degeneracy, uniform positive density, porosity and convergence of the free boundaries. Finally, we show that any solution to the limit operator is a limit of value functions for a specific Tug-of-War game.