Electron. J. Diff. Eqns., Vol. 2001(2001), No. 44, pp. 1-8.

An elementary proof of the Harnack inequality for non-negative infinity-superharmonic functions

Tilak Bhattacharya

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.

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Tilak Bhattacharya
Indian Statistical Institute
7, S.J.S. Sansanwal Marg
New Delhi 110 016 India
e-mail: tlk@isid.isid.ac.in

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