Geodesic
An elementary proof of the Harnack inequality for non-negative infinity-superharmonic functions
Electronic journal of differential equations, Tome 2001 (2001)
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.
Classification : 35J70, 26A16
Keywords: viscosity solutions, Harnack inequality, infinite harmonic operator, distance function 
@article{EJDE_2001__2001__a182,
     author = {Bhattacharya,  Tilak},
     title = {An elementary proof of the {Harnack} inequality for non-negative infinity-superharmonic functions},
     journal = {Electronic journal of differential equations},
     year = {2001},
     volume = {2001},
     zbl = {0966.35052},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2001__2001__a182/}
}
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Bhattacharya,  Tilak. An elementary proof of the Harnack inequality for non-negative infinity-superharmonic functions. Electronic journal of differential equations, Tome 2001 (2001). http://geodesic.mathdoc.fr/item/EJDE_2001__2001__a182/