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New Trends on Analysis and Geometry in Metric Spaces

Levico Terme, Italy 2017

By: Fabrice Baudoin,Séverine Rigot,Giuseppe Savaré,Nageswari Shanmugalingam

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New Trends on Analysis and Geometry in Metric Spaces

Levico Terme, Italy 2017

By: Fabrice Baudoin,Séverine Rigot,Giuseppe Savaré,Nageswari Shanmugalingam

This book includes four courses on geometric measure theory, the calculus of variations, partial differential equations, and differential geometry. Authored by leading experts in their fields, the lectures present different approaches to research topics with the common background of a relevant underlying, usually non-Riemannian, geometric structure. In particular, the topics covered concern differentiation and functions of bounded variation in metric spaces, Sobolev spaces, and differential geometry in the so-called Carnot–Carathéodory spaces. The text is based on lectures presented at the 10th School on "Analysis and Geometry in Metric Spaces" held in Levico Terme (TN), Italy, in collaboration with the University of Trento, Fondazione Bruno Kessler and CIME, Italy. The book is addressed to both graduate students and researchers.

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Page Count

312

Publisher

Springer Nature

Published Date

ISBN 10

3030841413

ISBN 13

9783030841416

Sobolev Spaces on Metric Measure Spaces

An Approach Based on Upper Gradients

By: Juha Heinonen,Pekka Koskela,Nageswari Shanmugalingam,Jeremy T. Tyson

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Sobolev Spaces on Metric Measure Spaces

An Approach Based on Upper Gradients

By: Juha Heinonen,Pekka Koskela,Nageswari Shanmugalingam,Jeremy T. Tyson

Analysis on metric spaces emerged in the 1990s as an independent research field providing a unified treatment of first-order analysis in diverse and potentially nonsmooth settings. Based on the fundamental concept of upper gradient, the notion of a Sobolev function was formulated in the setting of metric measure spaces supporting a Poincaré inequality. This coherent treatment from first principles is an ideal introduction to the subject for graduate students and a useful reference for experts. It presents the foundations of the theory of such first-order Sobolev spaces, then explores geometric implications of the critical Poincaré inequality, and indicates numerous examples of spaces satisfying this axiom. A distinguishing feature of the book is its focus on vector-valued Sobolev spaces. The final chapters include proofs of several landmark theorems, including Cheeger's stability theorem for Poincaré inequalities under Gromov–Hausdorff convergence, and the Keith–Zhong self-improvement theorem for Poincaré inequalities.

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Page Count

447

Publisher

Cambridge University Press

Published Date

ISBN 10

1316241033

ISBN 13

9781316241035