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