On the Properties of Infinity-Harmonic Functions and an Application to Capacitary Convex Rings

Abstract

We study positive ∞-harmonic functions in bounded domains. We use the theory of viscosity solutions in this work. We prove a boundary Harnack inequality and a comparison result for such functions near a flat portion of the boundary where they vanish. We also study ∞-capacitary functions on convex rings. We show that the gradient satisfies a global maximum principle, it is nonvanishing outside a set of measure zero and the level sets are star-shaped.