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研究生： 洪偉翔
Hung, Wei-Hsiang
論文名稱： 完備流形上的波松方程
A note on Poisson equation on complete manifolds
指導教授： 宋瓊珠
Sung, Chiung-Jue
口試委員: 邱鴻麟
Chiu, Hung-Lin
蕭育如
Syau, Yu-Ru
學位類別： 碩士
Master
系所名稱： 理學院 - 數學系
Department of Mathematics
論文出版年： 2019
畢業學年度： 107
語文別： 英文
論文頁數： 97
中文關鍵詞： 加權的龐加萊不等式 、熱核 、格林函數 、波松方程
外文關鍵詞： weighted Poincare inequality, heat kernel, Green's function, Poisson equation
相關次數： 點閱：2  下載：0 
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    • 我們首先研究熱核和格林函數在Bakry-Emery里奇曲率有下界且加權函數的振幅有上界的平滑度量測度空間的估計。當完備流形滿足加權的龐加萊不等式而且此加權函數有下界，我們得到熱核和格林函數的估計，並且利用這些估計，我們得到波松方程解的存在性及其估計。

    Let $(M, g, e^{-f}dx)$ be a smooth metric measure space with the associated Bakry-Emery Ricci curvature bounded below. Assume that the oscillation of weighted function is uniformly bounded from above on any unit ball and weighted function $\rho$ is bounded from below, we study the heat kernel estimates and Green’s function estimates on $M$ if $M$ admits a weighted Poincar$\acute{e}$ inequality. As an application, we then estimate the solution of the Poisson equation on such $M$.

    1. Introduction-3 2. Preliminaries-7 2.1. Heat kerne-l8 2.2. Green's function-10 2.3. Gradient estimate-16 3. Estimates for the heat kernel-19 4. Estimates for Green's function-38 5. Estimate for a solution of the Poisson equation-78 References-96

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    [9] O. Munteanu, C. J. Sung, and J. Wang, Poisson equation on complete manifolds, Adv. Math. 348 (2019), 81-145.

    [10] O. Munteanu, C. J. Sung, and J. Wang, Weighted Poincar e inequality and the Poisson equation, arXiv: 1904.13337v1 [math.DG] 30 Apr. 2019.

    [11] O. Munteanu and N. Sesum, The Poisson equation on complete manifolds with positive spectrum and applications, Adv. Math. 223 (2010), 198-219.

    [12] O. Munteanu and J. Wang, Smooth metric measure space with nonnegative curvature, Comm. Anal. Geom. 19 (2011), 451-486.

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