A convergent difference scheme for the infinity Laplacian: construction of absolutely minimizing Lipschitz extensions

Abstract

This article considers the problem of building absolutely minimizing Lipschitz extensions to a given function. These extensions can be characterized as being the solution of a degenerate elliptic partial differential equation, the “infinity Laplacian”, for which there exist unique viscosity solutions. A convergent difference scheme for the infinity Laplacian equation is introduced, which arises by minimizing the discrete Lipschitz constant of the solution at every grid point. Existence and uniqueness of solutions to the scheme is shown directly. Solutions are also shown to satisfy a discrete comparison principle. Solutions are computed using an explicit iterative scheme which is equivalent to solving the parabolic version of the equation.