On the differential structure of metric measure spaces and applications
著者
書誌事項
On the differential structure of metric measure spaces and applications
（Memoirs of the American Mathematical Society, no. 1113）
American Mathematical Society, 2015, c2014
大学図書館所蔵 件 / 全9件
-
該当する所蔵館はありません
- すべての絞り込み条件を解除する
注記
"Volume 236, number 1113 (third of 6 numbers), July 2015"
Includes bibliographical references (p. 89-91)
内容説明・目次
内容説明
The main goals of this paper are:
(i) To develop an abstract differential calculus on metric measure spaces by investigating the duality relations between differentials and gradients of Sobolev functions. This will be achieved without calling into play any sort of analysis in charts, our assumptions being: the metric space is complete and separable and the measure is Radon and non-negative.
(ii) To employ these notions of calculus to provide, via integration by parts, a general definition of distributional Laplacian, thus giving a meaning to an expression like $\Delta g=\mu$, where $g$ is a function and $\mu$ is a measure.
(iii) To show that on spaces with Ricci curvature bounded from below and dimension bounded from above, the Laplacian of the distance function is always a measure and that this measure has the standard sharp comparison properties. This result requires an additional assumption on the space, which reduces to strict convexity of the norm in the case of smooth Finsler structures and is always satisfied on spaces with linear Laplacian, a situation which is analyzed in detail.
目次
Introduction
Preliminaries
Differentials and gradients
Laplacian Comparison estimates
Appendix A. On the duality between cotangent and tangent spaces
Appendix B. Remarks about the definition of the Sobolev classes
References
「Nielsen BookData」 より
