加权Laplace在Bakry-Émery Ricci曲率条件下的Li-Yau梯度估计——关于Li-Yau梯度估计的研究
Li-Yau Gradient Estimation of Weighted Laplace under Bakry-Émery Ricci Curvature—Research on Li-Yau Gradient Estimation
摘要: 本文研究了在Bakry-Émery Ricci曲率条件下加权Laplace算子的Li-Yau梯度估计的问题，利用Bochner公式与加权Laplace公式以及极大值定理等处理Li-Yau梯度问题的方法，获得了加权Laplace在Bakry-Émery Ricci曲率有下界的条件下，热方程的正解
u (
x,
t)的最优Li-Yau梯度估计。
Abstract: In this paper, the problem of Li-Yau gradient estimation of weighted Laplace operator under Bakry-Émery Ricci curvature is studied. Bochner formula, weighted Laplace formula and the maximum theorem are used to deal with the Li-Yau gradient problem. The optimal Li-Yau gradient estimation for the positive solution u (x, t) of the heat equation is obtained under the condition of lower bound for weighted Laplace Bakry-Émery Ricci curvature.
参考文献
|
[1]
|
Li, P. and Yau, S.T. (1986) On the Parabolic Kernel of the Schrödinger Operator. Acta Mathematica, 156, 153-201. [Google Scholar] [CrossRef]
|
|
[2]
|
Davies, E.B. (1989) Heat Kernels and Spectral Theory. Cambridge Tracts in Math., Vol. 92, Cambridge University. Press.
|
|
[3]
|
Hamilton, R.S. (1993) Matrix Harnack Estimate for the Heat Equation. Communications in Analysis and Geometry, 1, 113-126. [Google Scholar] [CrossRef]
|
|
[4]
|
Bakry, D. and Qian, Z. (2005) Volume Comparison Theorems without Jacobi Fields. Conference on Potential Theory, Bucharest, 115-122.
|
|
[5]
|
Li, J. and Xu, X. (2011) Differential Harnack Inequalities on Riemannian Manifolds I: Linear Heat Equation. Advances in Mathematics, 226, 4456-4491. [Google Scholar] [CrossRef]
|
|
[6]
|
Qi, S. and Zhang, Q.S. (2021) A Sharp Li-Yau Gradient Bound on Compact Manifolds.
|
|
[7]
|
Song, X., Wu, L. and Zhu, M. (2024) A Direct Approach to Sharp Li-Yau Estimates on Closed Manifolds with Negative Ricci Lower Bound. Proceedings of the American Mathematical Society, 153, 291-305. [Google Scholar] [CrossRef]
|
|
[8]
|
Wu, J. (2024) An Improvement of the Sharp Li-Yau Bound on Closed Manifolds. Archiv der Mathematik, 123, 309-318. [Google Scholar] [CrossRef]
|
|
[9]
|
Song, X.Y. and Wu, L. (2023) Li-Yau Gradient Estimates on Closed Manifolds under Bakry-Mery Ricci Curvature Conditions. Journal of the Korean Mathematical Society, 60, 19.
|