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...
Gespeichert in:
| 1. Verfasser: | |
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| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Cambridge
Cambridge University Press
2015
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| Schriftenreihe: | New mathematical monographs
27 |
| Schlagworte: | |
| Online-Zugang: | BSB01 FHN01 Volltext |
| Zusammenfassung: | 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 |
| Beschreibung: | Title from publisher's bibliographic system (viewed on 05 Oct 2015) |
| Beschreibung: | 1 online resource (xii, 434 pages) |
| ISBN: | 9781316135914 |
| DOI: | 10.1017/CBO9781316135914 |
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| 245 | 1 | 0 | |a Sobolev spaces on metric measure spaces |b an approach based on upper gradients |c Juha Heinonen, Pekka Koskela, Nageswari Shanmugalingam, Jeremy T. Tyson |
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| 500 | |a Title from publisher's bibliographic system (viewed on 05 Oct 2015) | ||
| 505 | 8 | |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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| 700 | 1 | |a Shanmugalingam, Nageswari |e Sonstige |4 oth | |
| 700 | 1 | |a Tyson, Jeremy T. |d 1972- |e Sonstige |4 oth | |
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Datensatz im Suchindex
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| author | Heinonen, Juha |
| author_facet | Heinonen, Juha |
| author_role | aut |
| author_sort | Heinonen, Juha |
| author_variant | j h jh |
| building | Verbundindex |
| bvnumber | BV043940262 |
| classification_rvk | SK 600 |
| collection | ZDB-20-CBO |
| 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 | (ZDB-20-CBO)CR9781316135914 (OCoLC)992879659 (DE-599)BVBBV043940262 |
| dewey-full | 515/.7 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.7 |
| dewey-search | 515/.7 |
| dewey-sort | 3515 17 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| doi_str_mv | 10.1017/CBO9781316135914 |
| format | Electronic eBook |
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| id | DE-604.BV043940262 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T07:39:13Z |
| institution | BVB |
| isbn | 9781316135914 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-029349232 |
| oclc_num | 992879659 |
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| owner | DE-12 DE-92 |
| owner_facet | DE-12 DE-92 |
| physical | 1 online resource (xii, 434 pages) |
| psigel | ZDB-20-CBO ZDB-20-CBO BSB_PDA_CBO ZDB-20-CBO FHN_PDA_CBO |
| publishDate | 2015 |
| publishDateSearch | 2015 |
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| publisher | Cambridge University Press |
| record_format | marc |
| series2 | New mathematical monographs |
| spelling | Heinonen, Juha Verfasser aut 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 resource (xii, 434 pages) txt rdacontent c rdamedia cr rdacarrier New mathematical monographs 27 Title from publisher's bibliographic system (viewed on 05 Oct 2015) 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 Sobolev spaces Metrischer Raum (DE-588)4169745-5 gnd rswk-swf Sobolev-Raum (DE-588)4055345-0 gnd rswk-swf Sobolev-Raum (DE-588)4055345-0 s Metrischer Raum (DE-588)4169745-5 s 1\p DE-604 Koskela, Pekka Sonstige oth Shanmugalingam, Nageswari Sonstige oth Tyson, Jeremy T. 1972- Sonstige oth Erscheint auch als Druckausgabe 978-1-107-09234-1 Erscheint auch als Druckausgabe 978-1-107-46534-3 https://doi.org/10.1017/CBO9781316135914 Verlag URL des Erstveröffentlichers Volltext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Heinonen, Juha Sobolev spaces on metric measure spaces an approach based on upper gradients 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 Sobolev spaces Metrischer Raum (DE-588)4169745-5 gnd Sobolev-Raum (DE-588)4055345-0 gnd |
| subject_GND | (DE-588)4169745-5 (DE-588)4055345-0 |
| 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 Sobolev spaces Metrischer Raum (DE-588)4169745-5 gnd Sobolev-Raum (DE-588)4055345-0 gnd |
| topic_facet | Metric spaces Sobolev spaces Metrischer Raum Sobolev-Raum |
| url | https://doi.org/10.1017/CBO9781316135914 |
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