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

Brett Kotschwar, Lei Ni|ArXiv.org|Nov 14, 2007

Geometric Analysis and Curvature Flows8 references3 citations

TL;DR

This paper establishes local interior and boundary gradient estimates for p-harmonic functions on general Riemannian manifolds, leveraging a nonlinear Bochner-type formula and gradient estimate techniques. It applies these estimates to prove existence of weak solutions to the 1/H-flow (inverse mean curvature flow) under sharp volume growth assumptions and derives a sharp Li-Yau type gradient estimate and entropy monotonicity formula for parabolic p-Laplace equations on nonnegatively Ricci curved manifolds, implying Euclidean rigidity under sharp L^p-logarithmic Sobolev inequalities.

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-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.

Motivation & Objective

• To derive local interior and boundary gradient estimates for p-harmonic functions on general Riemannian manifolds with curvature bounds.
• To extend Moser’s strategy for 1/H-flow by establishing uniform gradient estimates as p→1, enabling existence of weak solutions under sharp volume growth conditions.
• To prove sharp Li-Yau type gradient estimates for positive solutions to two parabolic p-Laplace equations on manifolds with nonnegative Ricci curvature.
• To establish a monotonicity formula for an entropy functional, generalizing the linear heat equation case.
• To show that a complete Riemannian manifold with nonnegative Ricci curvature and a sharp L^p-logarithmic Sobolev inequality must be isometric to Euclidean space.

Proposed method

• Derive a nonlinear Bochner-type identity relating the p-Laplacian and its linearization to control the Hessian and gradient of p-harmonic functions.
• Apply the gradient estimate technique of Chen-Yan and Li-Yau to derive local interior estimates for |∇u|², where u = -(p−1)log v and v is p-harmonic.
• Construct local barrier functions to obtain boundary gradient estimates under a sharp volume growth assumption on the ambient manifold.
• Use the transformation v ↦ u = -(p−1)log v to relate p-harmonic functions to solutions of the 1/H-flow equation in the level-set formulation.
• Prove a sharp Li-Yau gradient estimate for positive solutions to the parabolic p-Laplace equations on manifolds with Ric ≥ 0.
• Derive an entropy monotonicity formula for one of the parabolic equations, generalizing the second author’s earlier result for the linear heat equation.

Experimental results

Research questions

• RQ1 Can local interior and boundary gradient estimates for p-harmonic functions be established on general Riemannian manifolds with curvature bounds, independent of Euclidean structure?
• RQ2 Does the Moser-type regularization strategy for 1/H-flow via p-harmonic functions yield weak solutions under sharp volume growth assumptions on the ambient manifold?
• RQ3 What are the sharp Li-Yau type gradient estimates for positive solutions to parabolic p-Laplace equations on manifolds with nonnegative Ricci curvature?
• RQ4 Is there a monotonic entropy formula for parabolic p-Laplace equations that generalizes the linear heat equation case?
• RQ5 Does a complete Riemannian manifold with nonnegative Ricci curvature and a sharp L^p-logarithmic Sobolev inequality necessarily have to be isometric to Euclidean space?

Key findings

• A local interior gradient estimate for |∇u|² is established on balls of radius R, with a bound that remains finite as p→1, under sectional curvature K_M ≥ -K².
• The bound is of the form sup_{B(x₀,R/2)} |∇u|² ≤ C(n,K,p,R,ε) with explicit dependence on n, K, p, R, and ε, and the constant remains finite as p→1.
• A boundary gradient estimate |∇u| ≤ H₊ + ε is obtained for p ≤ p₀(ε), enabling convergence of solutions to the 1/H-flow as p↘1.
• A sharp Li-Yau type gradient estimate is proven for positive solutions to two parabolic p-Laplace equations on manifolds with Ric ≥ 0, with explicit dependence on p and curvature.
• An entropy monotonicity formula is derived for one of the parabolic equations, generalizing the second author’s earlier result for the linear heat equation.
• The entropy formula implies that a complete Riemannian manifold with nonnegative Ricci curvature and a sharp L^p-logarithmic Sobolev inequality must be isometric to Euclidean space.