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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| Main Authors: | , , , |
|---|---|
| Format: | Electronic eBook |
| Language: | English |
| Published: |
Cambridge :
Cambridge University Press,
2015.
|
| Series: | New mathematical monographs ;
27. |
| Subjects: | |
| Online Access: | DE-862 DE-863 |
| 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 resource (xii, 434 pages) |
| Bibliography: | Includes bibliographical references and indexes. |
| ISBN: | 9781316248607 1316248607 9781316250495 1316250490 9781316135914 1316135918 |
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| 505 | 0 | |a 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. | |
| 520 | |a 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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| author | Heinonen, Juha Heinonen, Juha Koskela, Pekka Shanmugalingam, Nageswari Tyson, Jeremy T., 1972- |
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| author_facet | Heinonen, Juha Heinonen, Juha Koskela, Pekka Shanmugalingam, Nageswari Tyson, Jeremy T., 1972- |
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| 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. |
| ctrlnum | (OCoLC)900943780 |
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| discipline | Mathematik |
| format | Electronic eBook |
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| owner_facet | MAIN DE-862 DE-BY-FWS DE-863 DE-BY-FWS |
| physical | 1 online resource (xii, 434 pages) |
| psigel | ZDB-4-EBA FWS_PDA_EBA ZDB-4-EBA |
| publishDate | 2015 |
| publishDateSearch | 2015 |
| publishDateSort | 2015 |
| publisher | Cambridge University Press, |
| record_format | marc |
| series | New mathematical monographs ; |
| series2 | New mathematical monographs ; |
| spelling | Heinonen, Juha, author. http://id.loc.gov/authorities/names/n92086213 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. ©2015 1 online resource (xii, 434 pages) text txt rdacontent computer c rdamedia online resource cr rdacarrier New mathematical monographs ; 27 Includes bibliographical references and indexes. Print version record. 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. 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. Metric spaces. http://id.loc.gov/authorities/subjects/sh85084441 Sobolev spaces. http://id.loc.gov/authorities/subjects/sh85123836 Espaces métriques. Espaces de Sobolev. MATHEMATICS Calculus. bisacsh MATHEMATICS Mathematical Analysis. bisacsh Sobolev, Espacios de embucm Metric spaces fast Sobolev spaces fast Sobolev-Raum gnd http://d-nb.info/gnd/4055345-0 Metrischer Raum gnd http://d-nb.info/gnd/4169745-5 Electronic books. Koskela, Pekka, author. http://id.loc.gov/authorities/names/n00000963 Shanmugalingam, Nageswari, author. http://id.loc.gov/authorities/names/n2014041412 Tyson, Jeremy T., 1972- author. https://id.oclc.org/worldcat/entity/E39PCjtWqkhYMKwpDX7WRfb6JC http://id.loc.gov/authorities/names/n2010024742 has work: Sobolev spaces on metric measure spaces (Text) https://id.oclc.org/worldcat/entity/E39PCFJtdFHXqHBbXWFwY86btX https://id.oclc.org/worldcat/ontology/hasWork Print version: Heinonen, Juha. Sobolev spaces on metric measure spaces. Cambridge, United Kingdom : Cambridge University Press, 2015 9781107092341 (DLC) 2014027794 (OCoLC)883836458 New mathematical monographs ; 27. http://id.loc.gov/authorities/names/n2003010567 |
| spellingShingle | Heinonen, Juha Heinonen, Juha Koskela, Pekka Shanmugalingam, Nageswari Tyson, Jeremy T., 1972- Sobolev spaces on metric measure spaces : an approach based on upper gradients / New mathematical monographs ; 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. Metric spaces. http://id.loc.gov/authorities/subjects/sh85084441 Sobolev spaces. http://id.loc.gov/authorities/subjects/sh85123836 Espaces métriques. Espaces de Sobolev. MATHEMATICS Calculus. bisacsh MATHEMATICS Mathematical Analysis. bisacsh Sobolev, Espacios de embucm Metric spaces fast Sobolev spaces fast Sobolev-Raum gnd http://d-nb.info/gnd/4055345-0 Metrischer Raum gnd http://d-nb.info/gnd/4169745-5 |
| subject_GND | http://id.loc.gov/authorities/subjects/sh85084441 http://id.loc.gov/authorities/subjects/sh85123836 http://d-nb.info/gnd/4055345-0 http://d-nb.info/gnd/4169745-5 |
| 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 / |
| topic | Metric spaces. http://id.loc.gov/authorities/subjects/sh85084441 Sobolev spaces. http://id.loc.gov/authorities/subjects/sh85123836 Espaces métriques. Espaces de Sobolev. MATHEMATICS Calculus. bisacsh MATHEMATICS Mathematical Analysis. bisacsh Sobolev, Espacios de embucm Metric spaces fast Sobolev spaces fast Sobolev-Raum gnd http://d-nb.info/gnd/4055345-0 Metrischer Raum gnd http://d-nb.info/gnd/4169745-5 |
| topic_facet | Metric spaces. Sobolev spaces. Espaces métriques. Espaces de Sobolev. MATHEMATICS Calculus. MATHEMATICS Mathematical Analysis. Sobolev, Espacios de Metric spaces Sobolev spaces Sobolev-Raum Metrischer Raum Electronic books. |
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