Electronic Journal of Differential Equations,
Vol. 2001(2001), No. 44, pp. 1-8.
Title: An elementary proof of the Harnack inequality for
non-negative infinity-superharmonic functions
Author: Tilak Bhattacharya (Indian Statistical Institute, New Delhi, India)
Abstract:
We present an elementary proof of the Harnack inequality for non-negative
viscosity supersolutions of $\Delta_{\infty}u=0$. This was originally proven
by Lindqvist and Manfredi using sequences of solutions of the $p$-Laplacian.
We work directly with the $\Delta_{\infty}$ 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.
Submitted January 15, 2001. Revised May 17, 2001. Published June 14, 2001.
Math Subject Classifications: 35J70, 26A16.
Key Words: Viscosity solutions; Harnack inequality; infinite harmonic operator;
distance function.