An infinity Laplace equation with gradient term and mixed boundary conditions

Abstract

Abstract. We obtain existence, uniqueness, and stability results for the modified 1-homogeneous infinity Laplace equationsubject 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.

General existence of solutions to dynamic programming equations

“…Recently, the deterministic or stochastic game-theoretic approach to various nonlinear partial differential equations has attracted a lot of attention; see, for example, a variety of games in [12,13,22,23,9,4,21,1,7] and related topics in [15,20,19,24]. The results in the literature so far can be summarized in the following way.…”

A deterministic‐control‐based approach to fully nonlinear parabolic and elliptic equations

“…For many second-order elliptic and parabolic equations, the solution is the value function of an associated stochastic control problem (see, e.g., [25,26]). For the infinity Laplacian, the Dirichlet problem can be solved using a rather simple two-person stochastic game [39] (see also [1,2,8,23,34,40,46,47] for related work including extensions to evolution problems and the p-Laplacian).…”

An obstacle problem for Tug-of-War games

“…Recently, Peres, Schram, Sheffield and Wilson, [14], discovered the relationship between limits of value functions of Tug-of-War games and solutions to the infinity Laplacian. Also, Peres and Sheffield, [15], found a game whose values approximate solutions to the p−Laplacian, see also the work by Manfredi, Parvianen and Rossi, [9], [10], [11], [12], Bjorland, Caffarelli and Figalli, [5], by Armstrong, Smart and Somersille, [2], by Peres, Peté and Somersille, [13], and by Antunovíc, Peres, Sheffield and Somersille, [1].…”