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

In this manuscript we deal with regularity issues and the asymptotic behaviour (as $p\to \mathrm{\infty }$) of solutions for elliptic free boundary problems of $p-$Laplacian type ($2\le p<\mathrm{\infty }$): \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 $\mathrm{\Omega }$ is a regular domain. First, we prove the convergence as $p\to \mathrm{\infty }$ 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, $$\{\begin{array}{rcrcl}\max \{-{\mathrm{\Delta }}_{\mathrm{\infty }}{u}_{\mathrm{\infty }},-\mathrm{\mid }\mathrm{\nabla }{u}_{\mathrm{\infty }}\mathrm{\mid }+{\chi }_{\{u_{\mathrm{\infty }}>0\}}\}& =& 0& \text{in}& \mathrm{\Omega }\cap \{u_{\mathrm{\infty }}\ge 0\}\\ u_{\mathrm{\infty }}& =& g& \text{on}& \mathrm{\partial }\mathrm{\Omega }\mathrm{.}\end{array}$$ 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.