An Elementary Proof of the Harnack Inequality for Non-Negative Infinity-Superharmonic Functions

Abstract

We present an elementary proof of the Harnack inequality for non-negative viscosity supersolutions of Δ∞u = 0. This was originally proven by Lindqvist and Manfredi using sequences of solutions of the p-Laplacian. We work directly with the Δ∞ operator using the distance function as a test function. We also provide simple proofs of the Liouville property, Hopf boundary point lemma and Lipschitz continuity.