Sobolev spaces on metric measure spaces an approach based on upper gradients

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 m...

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Bibliographic Details
Main Authors: Heinonen, Juha, Koskela, Pekka (Author), Shanmugalingam, Nageswari (Author), Tyson, Jeremy T. (Author)
Format: eBook
Language:English
Published: Cambridge Cambridge University Press 2015
Series:New mathematical monographs
Subjects:
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Collection: Cambridge Books Online - Collection details see MPG.ReNa
Description
Summary: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
Physical Description:xii, 434 pages digital
ISBN:9781316135914