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.