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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| Other Authors: | , , |
| Format: | eBook |
| Language: | English |
| Published: |
Cambridge
Cambridge University Press
2015
|
| Series: | New mathematical monographs
27 |
| Links: | https://doi.org/10.1017/CBO9781316135914 |
| 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: | 1 Online-Ressource (xii, 434 Seiten) |
| ISBN: | 9781316135914 |
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| spelling | Heinonen, Juha Sobolev spaces on metric measure spaces an approach based on upper gradients Juha Heinonen, Pekka Koskela, Nageswari Shanmugalingam, Jeremy T. Tyson Cambridge Cambridge University Press 2015 1 Online-Ressource (xii, 434 Seiten) txt c cr New mathematical monographs 27 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. Koskela, Pekka Shanmugalingam, Nageswari Tyson, Jeremy T. 1972- Erscheint auch als Druck-Ausgabe 9781107092341 Erscheint auch als Druck-Ausgabe 9781107465343 |
| spellingShingle | Heinonen, Juha Sobolev spaces on metric measure spaces an approach based on upper gradients |
| title | Sobolev spaces on metric measure spaces an approach based on upper gradients |
| title_auth | Sobolev spaces on metric measure spaces an approach based on upper gradients |
| title_exact_search | Sobolev spaces on metric measure spaces an approach based on upper gradients |
| title_full | Sobolev spaces on metric measure spaces an approach based on upper gradients Juha Heinonen, Pekka Koskela, Nageswari Shanmugalingam, Jeremy T. Tyson |
| title_fullStr | Sobolev spaces on metric measure spaces an approach based on upper gradients Juha Heinonen, Pekka Koskela, Nageswari Shanmugalingam, Jeremy T. Tyson |
| title_full_unstemmed | Sobolev spaces on metric measure spaces an approach based on upper gradients Juha Heinonen, Pekka Koskela, Nageswari Shanmugalingam, Jeremy T. Tyson |
| title_short | Sobolev spaces on metric measure spaces |
| title_sort | sobolev spaces on metric measure spaces an approach based on upper gradients |
| title_sub | an approach based on upper gradients |
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