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 → ∞) of solutions for elliptic free boundary problems of p−Laplacian type (2 ≤ p < ∞): −pu(x) + λ0(x)χ{u>0}(x) = 0 in ⊂ RN , with a prescribed Dirichlet boundary data, where λ0 > 0 is a bounded function and is a regular domain. First, we prove the convergence as p → ∞ of any family of solutions (u p)p≥2, as well as we obtain the corresponding limit operator (in non-divergence form) ruling the limit equation, max −∞u∞, −|∇u∞| + χ{u∞>0} = 0 in ∩ {u∞ ≥ 0} u∞ = F on ∂ . Next, we obtain uniqueness for solutions to this limit problem. Finally, we show that any solution to the limit operator is a limit of value functions for a specific Tug-of-War game.