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In the first part of this paper, we prove local interior and boundarygradient estimates for p-harmonic functions on general Riemannian manifolds.With these estimates, following the strategy in recent work of R. Moser, weprove an existence theorem for weak solutions to the level set formulation ofthe 1/H (inverse mean curvature) flow for hypersurfaces in ambient manifoldssatisfying a sharp volume growth assumption. In the second part of this paper,we consider two parabolic analogues of the p-Laplace equation and prove sharpLi-Yau type gradient estimates for positive solutions to these equations onmanifolds of nonnegative Ricci curvature. For one of these equations, we alsoprove an entropy monotonicity formula generalizing an earlier such formula ofthe second author for the linear heat equation. As an application of thisformula, we show that a complete Riemannian manifold with non-negative Riccicurvature and sharp L^p-logarithmic Sobolev inequality must be isometric toEuclidean space.

Local gradient estimates of p-harmonic functions, 1/H-flow, and an entropy formula