| Description |
1 online resource (xii, 434 pages). |
| Physical Medium |
polychrome |
| Description |
text file |
| Series |
New mathematical monographs ; 27
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|
New mathematical monographs ; 27.
|
| Bibliography |
Includes bibliographical references and indexes. |
| Contents |
Introduction -- Review of basic functional analysis -- Lebesgue theory of Banach space-valued functions -- Lipschitz functions and embeddings -- Path integrals and modulus -- Upper gradients -- Sobolev spaces -- Poincaré inequalities -- Consequences of Poincaré inequalities -- Other definitions of Sobolev-type spaces -- Gromov-Hausdorff convergence and Poincaré inequalities -- Self-improvement of Poincaré inequalities -- An introduction to Cheeger's differentiation theory -- Examples, applications, and further research directions. |
| 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. |
| Local Note |
eBooks on EBSCOhost EBSCO eBook Subscription Academic Collection - North America |
| Subject |
Metric spaces.
|
|
Metric spaces. |
|
Sobolev spaces.
|
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Sobolev spaces. |
| Genre/Form |
Electronic books.
|
| Added Author |
Heinonen, Juha, author.
|
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Koskela, Pekka, author.
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Shanmugalingam, Nageswari, author.
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Tyson, Jeremy T., 1972- author.
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| Other Form: |
Print version: Heinonen, Juha. Sobolev spaces on metric measure spaces. Cambridge, United Kingdom : Cambridge University Press, 2015 9781107092341 (DLC) 2014027794 (OCoLC)883836458 |
| ISBN |
9781316248607 (electronic book) |
|
1316248607 (electronic book) |
|
9781316250495 (electronic book) |
|
1316250490 (electronic book) |
|
9781316135914 |
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1316135918 |
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9781107092341 (hardback) |
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1107092345 (hardback) |
|
