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

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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Published 3 February 2012