Discontinuous Gradient Constraints and the Infinity Laplacian

Motivated by tug- of-war games and asymptotic analysis of certain variational problems, we consider the following gradient constraint problem: given a bounded domain Omega subset of R-n, a continuous function f : partial derivative Omega -> R, and a nonempty subset D subset of Omega, find a solution to {min {Delta(infinity)u,vertical bar Du vertical bar - chi(D)} = 0 in Omega u = f on partial derivative Omega, where Delta(infinity) is the infinity Laplace operator. We prove that this problem always has a solution that is unique if (D) over bar = (intD) over bar. If this regularity condition on D fails, then solutions obtained from game theory and L-p-approximation need not coincide.