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

A VERSION OF THE {H}OPF-{L}AX FORMULA IN THE {H} EISENBERGGROUP

We consider Hamilton-Jacobi equations in the Heisenberg group . We establish uniqueness of bounded viscosity solutions with continuous initial data . When the hamiltonian H is radial, convex and superlinear the solution is given by the Hopf-Lax formula where the Lagrangian L is the horizontal Legendre transform of H lifted to the algebra by requiring it to be radial with respect to the Carnot-Carathéodory metric