Variational p-harmonious functions: existence and convergence to p-harmonic functions

In a recent paper, the last three authors showed that a game-theoretic p-harmonic function u is characterized by an asymptotic mean-value property with respect to a kind of nonlinear mean value m(u)(x), defined variationally on a ball of radius r centered at x. In this paper, we consider the operator acting on continuous functions u on in a domain of the Euclidean space. This is defined by suitably modifying m(u)(x). We first derive various properties of this operator such as continuity and monotonicity. Then, we prove the existence and uniqueness of a function u satisfying the Dirichlet-type problem:u_r(x)=m(u_r)(x) for every x in the domain, for any given continuous boundary values.This result holds, if we assume the existence of a suitable notion of barrier for all points of the boundary. We call this u the variational p-harmonious function with given Dirichlet boundary data. It is obtained by means of a Perron-type method based on a comparison principle. We then show that the family {u_r} gives a uniform approximation for the viscosity solution of the p-Laplace equation, with the same boundary data.