
On the Differential Structure of Metric Measure Spaces and Applications
Nicola Gigli(Author)
American Mathematical Society (Publisher)
Will be published approx. on 30. July 2015
Book
Paperback/Softback
91 pages
978-1-4704-1420-7 (ISBN)
Description
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.
(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.
More details
Series
Language
English
Place of publication
Providence
United States
Target group
Professional and scholarly
Dimensions
Height: 254 mm
Width: 178 mm
Weight
180 gr
ISBN-13
978-1-4704-1420-7 (9781470414207)
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Schweitzer Classification
Person
Nicola Gigli, University of Bordeaux 1, France.
Content
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
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