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

Abstract

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-harmonic 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 nonnegative Ricci curvature and sharp L p -logarithmic Sobolev inequality must be isometric to Euclidean space.

Weak solutions of inverse mean curvature flow for hypersurfaces with boundary

“…with the help of p-harmonic functions. Later, Kotschwar and Ni [16] extended this result to Riemannian ambient spaces satisfying a volume growth condition.…”

Perelman’s entropy formula for the Witten Laplacian on Riemannian manifolds via Bakry–Emery Ricci curvature

“…See also [7,21,27]. Since Perelman's preprint [33] was published on Arxiv in 2002, many people have extended the monotonicity of the W-entropy to other type geometric heat flows on Riemannian manifolds [13,20,25,29,30]. In [29,30], Ni studied the W-entropy for the linear heat equation on complete Riemannian manifolds.…”

Liouville properties for 𝑝-harmonic maps with finite 𝑞-energy

“…Then, 0 ≤ h i ≤ 1, and by gradient estimate ( [21]), there are subsequence, say {h i } , converges, locally uniformly, to a weakly p-harmonic function h on M, satisfying 0 ≤ h ≤ 1. On E 1 , the maximum principle implies 1 − u…”

“…In [21], B. Kotschwar and L. Ni use a Bochner's formula on a neighborhood of the maximum point (i.e. the p-Laplace operator is neither degenerate nor singular elliptic on this neighborhood) to prove a gradient estimate for positive p-harmonic functions.…”

“…We also note that, the perturbation method we employed in studying the p-Laplace equation is in contrast to the methods in [41] for harmonic maps on surfaces, in [8] for the level-set formulation of the mean curvature flow, in [17] for the inverse mean curvature flow, and in [21] for certain parabolic equations associated to the p-Laplacian. Theorem 1.1.…”