在完備黎曼流形上針對熱方程的哈納克不等式

Abstract

設 M 是一個光滑且連通的完備非緊緻黎曼流形. 若 M 滿足體積倍增條件和弱 L2 龐加萊不等式的話, 則針對底律雷特熱方程正解的哈納克不等式成立. 本論文主要探討這個定理. 基本上, 本論文可分成四個部分. 第一部分討論具有體積倍增條件和弱 L2 龐加萊不等式的流形上的一些重要性質. 第二部分則利用這些性質證明納許不等式和索伯列夫不等式. 第三部分著重在底律雷特熱方程的 subsolutions 和 supersolutions 並分別從這兩種類型的解中萃取出均值不等式及逆赫爾德不等式. 最後一部分則運用了在前三個部份中所獲得的工具來完成本論文主要定理的證明.

Let M be a smooth connected complete non-compact Riemannian manifold. If M satisfies the volume doubling condition (VDC) and the weak L2 Poincaré inequality (WPI), then the Harnack inequality for positive solutions to the Dirchlet heat equation holds on M. This is the main theorem in this paper. Basically, This paper can be seperated into four part. The first part discusses some important and useful properties on the manifold equipped with both VDC and WPI. The second part utilizes those properties to establish the Nash inequality and the Sobolev inequality. The third part focuses on subsolutions and supersolutions to the Dirichlet heat equation, and extracts the mean value inequality and the reverse Hölder inequality from them respectively. The last part applies all the tools obtained in previous parts to show the proof of the main theorem in this paper.