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About this product
- DescriptionThe 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 n-negative.*(ii) To employ these tions of calculus to provide, via integration by parts, a general definition of distributional Laplacian, thus giving a meaning to an expression like $Þlta g=\mu$, 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 rm in the case of smooth Finsler structures and is always satisfied on spaces with linear Laplacian, a situation which is analyzed in detail.
- Author BiographyNicola Gigli, University of Bordeaux 1, France.
- Author(s)Nicola Gigli
- PublisherAmerican Mathematical Society
- Date of Publication30/07/2015
- Series TitleMemoirs of the American Mathematical Society
- Series Part/Volume Number1113
- Place of PublicationProvidence
- Country of PublicationUnited States
- ImprintAmerican Mathematical Society
- Weight525 g
- Width178 mm
- Height254 mm
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