Sobolev spaces on metric measure spaces : an approach based on upper gradients / 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 m...
Uniform Title: | New mathematical monographs ;
27. |
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Main Authors: | |
Language: | English |
Published: |
Cambridge :
Cambridge University Press,
2015.
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Series: | New mathematical monographs ;
27. |
Subjects: | |
Online Access: | |
Physical Description: | 1 online resource (xii, 434 pages) : digital, PDF file(s). |
Format: | Electronic eBook |
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. |
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Note: | Title from publisher's bibliographic system (viewed on 05 Oct 2015). |
Call Number: | QA611.28 .H45 2015 |
ISBN: | 9781316135914 (ebook) |
DOI: | 10.1017/CBO9781316135914 |