An obstacle problem for Tug-of-War games

We consider the obstacle problem for the infinity Laplace equation. Given a Lipschitz boundary function and a Lipschitz obstacle we prove the existence and uniqueness of a super infinity-harmonic function constrained to lie above the obstacle which is infinity harmonic where it lies strictly above the obstacle. Moreover, we show that this function is the limit of value functions of a game we call obstacle tug-of-war

An infinity Laplace equation with gradient term and mixed boundary conditions

We obtain existence, uniqueness, and stability results for the modified 1-homogeneous infinity Laplace equation $$-\Delta_\infty u - \beta |Du| = f,$$ subject to Dirichlet or mixed Dirichlet-Neumann boundary conditions. Our arguments rely on comparing solutions of the PDE to subsolutions and supersolutions of a certain finite difference approximation.Comment: 13 pages, minor mistakes and typos correcte

An infinity Laplace equation with gradient term and mixed boundary conditions

Abstract. We obtain existence, uniqueness, and stability results for the modified 1-homogeneous infinity Laplace equation −Δ∞u − β|Du | = f, subject to Dirichlet or mixed Dirichlet-Neumann boundary conditions. Our arguments rely on comparing solutions of the PDE to subsolutions and supersolutions of a certain finite difference approximation. 1