Communications in Analysis and Geometry

Volume 19 (2011)

Number 4

Local gradient estimate for $p$-harmonic functions on Riemannian manifolds

Pages: 759 – 771

DOI: http://dx.doi.org/10.4310/CAG.2011.v19.n4.a4

Authors

Xiaodong Wang (Department of Mathematics, Michigan State University)

Lei Zhang (Department of Mathematics, University of Florida)

Abstract

For positive $p$-harmonic functions on Riemannian manifolds, we derive agradient estimate and Harnack inequality with constants depending only onthe lower bound of the Ricci curvature, the dimension $n$, $p$ and theradius of the ball on which the function is defined. Our approach is basedon a careful application of the Moser iteration technique and is differentfrom Cheng–Yau’s method employed by Kostchwar and Ni, inwhich a gradient estimate for positive $p$-harmonic functions is derivedunder the assumption that the sectional curvature is bounded from below.

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