The streamlines of ∞-harmonic functions obey the inverse mean curvature flow

Abstract

Given an ∞-harmonic function on a domain consider the function If with and then it is easy to check that the streamlines of are the level sets of w and w solves the level set formulation of the inverse mean curvature flow. For less regular solutions, neither statement is true in general, but even so, w is still a weak solution of the inverse mean curvature flow under far weaker assumptions. This is proved through an approximation of by p-harmonic functions, the use of conjugate -harmonic functions, and the known connection of the latter with the inverse mean curvature flow. A statement about the regularity of arises as a by-product.