p-调和函数、1/H-流和熵公式的局部梯度估计
Local gradient estimates of p-harmonic functions, 1/H-flow, and an entropy formula
摘要：
本文的第一部分证明了一般黎曼流形上p-调和函数的局部内部梯度和边界梯度估计。在此基础上，根据R. Moser，我们证明了一个关于满足体积急剧增长假设的超曲面的1/H（逆中曲率）流的水平集形式的弱解的存在性定理.在本文的第二部分中，我们考虑了p-Laplace方程的两个抛物型类似方程，并证明了这两个方程在非负里奇曲率流形上的正解的Li-Yau型梯度估计.对于其中一个方程，我们还证明了一个熵单调性公式，推广了第二作者的线性热方程的一个较早的这样的公式。作为这个公式的应用，我们证明了具有非负里奇曲率和尖锐的L ^p-对数Sobolev不等式的完备黎曼流形必等距于欧几里得空间.
英文摘要：
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.
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