Aspects of Sobolev-type inequalities:
Gespeichert in:
| Beteilige Person: | |
|---|---|
| Format: | Elektronisch E-Book |
| Sprache: | Englisch |
| Veröffentlicht: |
Cambridge
Cambridge University Press
2002
|
| Schriftenreihe: | London Mathematical Society lecture note series
289 |
| Schlagwörter: | |
| Links: | http://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&db=nlabk&AN=552338 http://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&db=nlabk&AN=552338 http://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&db=nlabk&AN=552338 |
| Beschreibung: | Includes bibliographical references (p. 183-188) and index 1 - Sobolev inequalities in R[superscript n] - 7 -- - 1.1 - Sobolev inequalities - 7 -- - 1.1.2 - The proof due to Gagliardo and to Nirenberg - 9 -- - 1.1.3 - p = 1 implies p [greater than or equal] 1 - 10 -- - 1.2 - Riesz potentials - 11 -- - 1.2.1 - Another approach to Sobolev inequalities - 11 -- - 1.2.2 - Marcinkiewicz interpolation theorem - 13 -- - 1.2.3 - Proof of Sobolev Theorem 1.2.1 - 16 -- - 1.3 - Best constants - 16 -- - 1.3.1 - The case p = 1: isoperimetry - 16 -- - 1.3.2 - A complete proof with best constant for p = 1 - 18 -- - 1.3.3 - The case p > 1 - 20 -- - 1.4 - Some other Sobolev inequalities - 21 -- - 1.4.1 - The case p > n - 21 -- - 1.4.2 - The case p = n - 24 -- - 1.4.3 - Higher derivatives - 26 -- - 1.5 - Sobolev -- Poincare inequalities on balls - 29 -- - 1.5.1 - The Neumann and Dirichlet eigenvalues - 29 -- - 1.5.2 - Poincare inequalities on Euclidean balls - 30 -- - 1.5.3 - Sobolev -- Poincare inequalities - 31 -- - 2 - Moser's elliptic Harnack inequality - 33 -- - 2.1 - Elliptic operators in divergence form - 33 -- - 2.1.1 - Divergence form - 33 -- - 2.1.2 - Uniform ellipticity - 34 -- - 2.1.3 - A Sobolev-type inequality for Moser's iteration - 37 -- - 2.2 - Subsolutions and supersolutions - 38 -- - 2.2.1 - Subsolutions - 38 -- - 2.2.2 - Supersolutions - 43 -- - 2.2.3 - An abstract lemma - 47 -- - 2.3 - Harnack inequalities and continuity - 49 -- - 2.3.1 - Harnack inequalities - 49 -- - 2.3.2 - Holder continuity - 50 -- - 3 - Sobolev inequalities on manifolds - 53 -- - 3.1.1 - Notation concerning Riemannian manifolds - 53 -- - 3.1.2 - Isoperimetry - 55 -- - 3.1.3 - Sobolev inequalities and volume growth - 57 -- - 3.2 - Weak and strong Sobolev inequalities - 60 -- - 3.2.1 - Examples of weak Sobolev inequalities - 60 -- - 3.2.2 - (S[superscript [theta] subscript r,s])-inequalities: the parameters q and v - 61 -- - 3.2.3 - The case 0 < q < [infinity] - 63 -- - 3.2.4 - The case 1 = [infinity] - 66 -- - 3.2.5 - The case -[infinity] < q < 0 - 68 -- - 3.2.6 - Increasing p - 70 -- - 3.2.7 - Local versions - 72 -- - 3.3.1 - Pseudo-Poincare inequalities - 73 -- - 3.3.2 - Pseudo-Poincare technique: local version - 75 -- - 3.3.3 - Lie groups - 77 -- - 3.3.4 - Pseudo-Poincare inequalities on Lie groups - 79 -- - 3.3.5 - Ricci [greater than or equal] 0 and maximal volume growth - 82 -- - 3.3.6 - Sobolev inequality in precompact regions - 85 -- - 4 - Two applications - 87 -- - 4.1 - Ultracontractivity - 87 -- - 4.1.1 - Nash inequality implies ultracontractivity - 87 -- - 4.1.2 - The converse - 91 -- - 4.2 - Gaussian heat kernel estimates - 93 -- - 4.2.1 - The Gaffney-Davies L[superscript 2] estimate - 93 -- - 4.2.2 - Complex interpolation - 95 -- - 4.2.3 - Pointwise Gaussian upper bounds - 98 -- - 4.2.4 - On-diagonal lower bounds - 99 -- - 4.3 - The Rozenblum-Lieb-Cwikel inequality - 103 -- - 4.3.1 - The Schrodinger operator [Delta] -- V - 103 -- - 4.3.2 - The operator T[subscript V] = [Delta superscript -1]V - 105 -- - 4.3.3 - The Birman-Schwinger principle - 109 -- - 5 - Parabolic Harnack inequalities - 111 -- - 5.1 - Scale-invariant Harnack principle - 111 -- - 5.2 - Local Sobolev inequalities - 113 -- - 5.2.1 - Local Sobolev inequalities and volume growth - 113 -- - 5.2.2 - Mean value inequalities for subsolutions - 119 -- - 5.2.3 - Localized heat kernel upper bounds - 122 -- - 5.2.4 - Time-derivative upper bounds - 127 -- - 5.2.5 - Mean value inequalities for supersolutions - 128 -- - 5.3 - Poincare inequalities - 130 -- - 5.3.1 - Poincare inequality and Sobolev inequality - 131 -- - 5.3.2 - Some weighted Poincare inequalities - 133 -- - 5.3.3 - Whitney-type coverings - 135 -- - 5.3.4 - A maximal inequality and an application - 139 -- - 5.3.5 - End of the proof of Theorem 5.3.4 - 141 -- - 5.4 - Harnack inequalities and applications - 143 -- - 5.4.1 - An inequality for log u - 143 -- - 5.4.2 - Harnack inequality for positive supersolutions - 145 -- - 5.4.3 - Harnack inequalities for positive solutions - 146 -- - 5.4.4 - Holder continuity - 149 -- - 5.4.5 - Liouville theorems - 151 -- - 5.4.6 - Heat kernel lower bounds - 152 -- - 5.4.7 - Two-sided heat kernel bounds - 154 -- - 5.5 - The parabolic Harnack principle - 155 -- - 5.5.1 - Poincare, doubling, and Harnack - 157 -- - 5.5.2 - Stochastic completeness - 161 -- - 5.5.3 - Local Sobolev inequalities and the heat equation - 164 -- - 5.5.4 - Selected applications of Theorem 5.5.1 - 168 -- - 5.6.1 - Unimodular Lie groups - 172 -- - 5.6.2 - Homogeneous spaces - 175 -- - 5.6.3 - Manifolds with Ricci curvature bounded below - 176 Focusing on Poincaré, Nash and other Sobolev-type inequalities and their applications to the Laplace and heat diffusion equations on Riemannian manifolds, this text is an advanced graduate book that will also suit researchers |
| Umfang: | 1 Online-Ressource (x, 190 p.) |
| ISBN: | 0521006074 1107360749 9780521006071 9781107360747 |
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| 245 | 1 | 0 | |a Aspects of Sobolev-type inequalities |c Laurent Saloff-Coste |
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| 490 | 0 | |a London Mathematical Society lecture note series |v 289 | |
| 500 | |a Includes bibliographical references (p. 183-188) and index | ||
| 500 | |a 1 - Sobolev inequalities in R[superscript n] - 7 -- - 1.1 - Sobolev inequalities - 7 -- - 1.1.2 - The proof due to Gagliardo and to Nirenberg - 9 -- - 1.1.3 - p = 1 implies p [greater than or equal] 1 - 10 -- - 1.2 - Riesz potentials - 11 -- - 1.2.1 - Another approach to Sobolev inequalities - 11 -- - 1.2.2 - Marcinkiewicz interpolation theorem - 13 -- - 1.2.3 - Proof of Sobolev Theorem 1.2.1 - 16 -- - 1.3 - Best constants - 16 -- - 1.3.1 - The case p = 1: isoperimetry - 16 -- - 1.3.2 - A complete proof with best constant for p = 1 - 18 -- - 1.3.3 - The case p > 1 - 20 -- - 1.4 - Some other Sobolev inequalities - 21 -- - 1.4.1 - The case p > n - 21 -- - 1.4.2 - The case p = n - 24 -- - 1.4.3 - Higher derivatives - 26 -- - 1.5 - Sobolev -- Poincare inequalities on balls - 29 -- - 1.5.1 - The Neumann and Dirichlet eigenvalues - 29 -- - 1.5.2 - Poincare inequalities on Euclidean balls - 30 -- - 1.5.3 - Sobolev -- Poincare inequalities - 31 -- - 2 - Moser's elliptic Harnack inequality | ||
| 500 | |a - 33 -- - 2.1 - Elliptic operators in divergence form - 33 -- - 2.1.1 - Divergence form - 33 -- - 2.1.2 - Uniform ellipticity - 34 -- - 2.1.3 - A Sobolev-type inequality for Moser's iteration - 37 -- - 2.2 - Subsolutions and supersolutions - 38 -- - 2.2.1 - Subsolutions - 38 -- - 2.2.2 - Supersolutions - 43 -- - 2.2.3 - An abstract lemma - 47 -- - 2.3 - Harnack inequalities and continuity - 49 -- - 2.3.1 - Harnack inequalities - 49 -- - 2.3.2 - Holder continuity - 50 -- - 3 - Sobolev inequalities on manifolds - 53 -- - 3.1.1 - Notation concerning Riemannian manifolds - 53 -- - 3.1.2 - Isoperimetry - 55 -- - 3.1.3 - Sobolev inequalities and volume growth - 57 -- - 3.2 - Weak and strong Sobolev inequalities - 60 -- - 3.2.1 - Examples of weak Sobolev inequalities - 60 -- - 3.2.2 - (S[superscript [theta] subscript r,s])-inequalities: the parameters q and v - 61 -- - 3.2.3 - The case 0 < q < [infinity] - 63 -- - 3.2.4 - The case 1 = [infinity] - 66 -- - 3.2.5 - The case -[infinity] < q < 0 | ||
| 500 | |a - 68 -- - 3.2.6 - Increasing p - 70 -- - 3.2.7 - Local versions - 72 -- - 3.3.1 - Pseudo-Poincare inequalities - 73 -- - 3.3.2 - Pseudo-Poincare technique: local version - 75 -- - 3.3.3 - Lie groups - 77 -- - 3.3.4 - Pseudo-Poincare inequalities on Lie groups - 79 -- - 3.3.5 - Ricci [greater than or equal] 0 and maximal volume growth - 82 -- - 3.3.6 - Sobolev inequality in precompact regions - 85 -- - 4 - Two applications - 87 -- - 4.1 - Ultracontractivity - 87 -- - 4.1.1 - Nash inequality implies ultracontractivity - 87 -- - 4.1.2 - The converse - 91 -- - 4.2 - Gaussian heat kernel estimates - 93 -- - 4.2.1 - The Gaffney-Davies L[superscript 2] estimate - 93 -- - 4.2.2 - Complex interpolation - 95 -- - 4.2.3 - Pointwise Gaussian upper bounds - 98 -- - 4.2.4 - On-diagonal lower bounds - 99 -- - 4.3 - The Rozenblum-Lieb-Cwikel inequality - 103 -- - 4.3.1 - The Schrodinger operator [Delta] -- V - 103 -- - 4.3.2 - The operator T[subscript V] = [Delta superscript -1]V - 105 -- - 4.3.3 | ||
| 500 | |a - The Birman-Schwinger principle - 109 -- - 5 - Parabolic Harnack inequalities - 111 -- - 5.1 - Scale-invariant Harnack principle - 111 -- - 5.2 - Local Sobolev inequalities - 113 -- - 5.2.1 - Local Sobolev inequalities and volume growth - 113 -- - 5.2.2 - Mean value inequalities for subsolutions - 119 -- - 5.2.3 - Localized heat kernel upper bounds - 122 -- - 5.2.4 - Time-derivative upper bounds - 127 -- - 5.2.5 - Mean value inequalities for supersolutions - 128 -- - 5.3 - Poincare inequalities - 130 -- - 5.3.1 - Poincare inequality and Sobolev inequality - 131 -- - 5.3.2 - Some weighted Poincare inequalities - 133 -- - 5.3.3 - Whitney-type coverings - 135 -- - 5.3.4 - A maximal inequality and an application - 139 -- - 5.3.5 - End of the proof of Theorem 5.3.4 - 141 -- - 5.4 - Harnack inequalities and applications - 143 -- - 5.4.1 - An inequality for log u - 143 -- - 5.4.2 - Harnack inequality for positive supersolutions - 145 -- - 5.4.3 - Harnack inequalities for positive solutions | ||
| 500 | |a - 146 -- - 5.4.4 - Holder continuity - 149 -- - 5.4.5 - Liouville theorems - 151 -- - 5.4.6 - Heat kernel lower bounds - 152 -- - 5.4.7 - Two-sided heat kernel bounds - 154 -- - 5.5 - The parabolic Harnack principle - 155 -- - 5.5.1 - Poincare, doubling, and Harnack - 157 -- - 5.5.2 - Stochastic completeness - 161 -- - 5.5.3 - Local Sobolev inequalities and the heat equation - 164 -- - 5.5.4 - Selected applications of Theorem 5.5.1 - 168 -- - 5.6.1 - Unimodular Lie groups - 172 -- - 5.6.2 - Homogeneous spaces - 175 -- - 5.6.3 - Manifolds with Ricci curvature bounded below - 176 | ||
| 500 | |a Focusing on Poincaré, Nash and other Sobolev-type inequalities and their applications to the Laplace and heat diffusion equations on Riemannian manifolds, this text is an advanced graduate book that will also suit researchers | ||
| 650 | 4 | |a Sobolev, Espaces de | |
| 650 | 4 | |a Inégalités (Mathématiques) | |
| 650 | 7 | |a Sobolev ruimten |2 gtt | |
| 650 | 7 | |a Vergelijkingen (wiskunde) |2 gtt | |
| 650 | 7 | |a Desigualdades (análise matemática) |2 larpcal | |
| 650 | 7 | |a MATHEMATICS / Functional Analysis |2 bisacsh | |
| 650 | 7 | |a Inequalities (Mathematics) |2 fast | |
| 650 | 7 | |a Sobolev spaces |2 fast | |
| 650 | 4 | |a Sobolev spaces | |
| 650 | 4 | |a Inequalities (Mathematics) | |
| 650 | 0 | 7 | |a Sobolevsche Ungleichung |0 (DE-588)4181715-1 |2 gnd |9 rswk-swf |
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Datensatz im Suchindex
| _version_ | 1818981175113285632 |
|---|---|
| any_adam_object | |
| author | Saloff-Coste, Laurent 1958- |
| author_GND | (DE-588)115395776 |
| author_facet | Saloff-Coste, Laurent 1958- |
| author_role | aut |
| author_sort | Saloff-Coste, Laurent 1958- |
| author_variant | l s c lsc |
| building | Verbundindex |
| bvnumber | BV043069374 |
| collection | ZDB-4-EBA |
| ctrlnum | (OCoLC)839304256 (DE-599)BVBBV043069374 |
| dewey-full | 515/.782 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.782 |
| dewey-search | 515/.782 |
| dewey-sort | 3515 3782 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Electronic eBook |
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68 -- - 3.2.6 - Increasing p - 70 -- - 3.2.7 - Local versions - 72 -- - 3.3.1 - Pseudo-Poincare inequalities - 73 -- - 3.3.2 - Pseudo-Poincare technique: local version - 75 -- - 3.3.3 - Lie groups - 77 -- - 3.3.4 - Pseudo-Poincare inequalities on Lie groups - 79 -- - 3.3.5 - Ricci [greater than or equal] 0 and maximal volume growth - 82 -- - 3.3.6 - Sobolev inequality in precompact regions - 85 -- - 4 - Two applications - 87 -- - 4.1 - Ultracontractivity - 87 -- - 4.1.1 - Nash inequality implies ultracontractivity - 87 -- - 4.1.2 - The converse - 91 -- - 4.2 - Gaussian heat kernel estimates - 93 -- - 4.2.1 - The Gaffney-Davies L[superscript 2] estimate - 93 -- - 4.2.2 - Complex interpolation - 95 -- - 4.2.3 - Pointwise Gaussian upper bounds - 98 -- - 4.2.4 - On-diagonal lower bounds - 99 -- - 4.3 - The Rozenblum-Lieb-Cwikel inequality - 103 -- - 4.3.1 - The Schrodinger operator [Delta] -- V - 103 -- - 4.3.2 - The operator T[subscript V] = [Delta superscript -1]V - 105 -- - 4.3.3</subfield></datafield><datafield tag="500" ind1=" " ind2=" "><subfield code="a"> - The Birman-Schwinger principle - 109 -- - 5 - Parabolic Harnack inequalities - 111 -- - 5.1 - Scale-invariant Harnack principle - 111 -- - 5.2 - Local Sobolev inequalities - 113 -- - 5.2.1 - Local Sobolev inequalities and volume growth - 113 -- - 5.2.2 - Mean value inequalities for subsolutions - 119 -- - 5.2.3 - Localized heat kernel upper bounds - 122 -- - 5.2.4 - Time-derivative upper bounds - 127 -- - 5.2.5 - Mean value inequalities for supersolutions - 128 -- - 5.3 - Poincare inequalities - 130 -- - 5.3.1 - Poincare inequality and Sobolev inequality - 131 -- - 5.3.2 - Some weighted Poincare inequalities - 133 -- - 5.3.3 - Whitney-type coverings - 135 -- - 5.3.4 - A maximal inequality and an application - 139 -- - 5.3.5 - End of the proof of Theorem 5.3.4 - 141 -- - 5.4 - Harnack inequalities and applications - 143 -- - 5.4.1 - An inequality for log u - 143 -- - 5.4.2 - Harnack inequality for positive supersolutions - 145 -- - 5.4.3 - Harnack inequalities for positive solutions</subfield></datafield><datafield tag="500" ind1=" " ind2=" "><subfield code="a"> - 146 -- - 5.4.4 - Holder continuity - 149 -- - 5.4.5 - Liouville theorems - 151 -- - 5.4.6 - Heat kernel lower bounds - 152 -- - 5.4.7 - Two-sided heat kernel bounds - 154 -- - 5.5 - The parabolic Harnack principle - 155 -- - 5.5.1 - Poincare, doubling, and Harnack - 157 -- - 5.5.2 - Stochastic completeness - 161 -- - 5.5.3 - Local Sobolev inequalities and the heat equation - 164 -- - 5.5.4 - Selected applications of Theorem 5.5.1 - 168 -- - 5.6.1 - Unimodular Lie groups - 172 -- - 5.6.2 - Homogeneous spaces - 175 -- - 5.6.3 - Manifolds with Ricci curvature bounded below - 176</subfield></datafield><datafield tag="500" ind1=" " ind2=" "><subfield code="a">Focusing on Poincaré, Nash and other Sobolev-type inequalities and their applications to the Laplace and heat diffusion equations on Riemannian manifolds, this text is an advanced graduate book that will also suit researchers</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Sobolev, Espaces de</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Inégalités (Mathématiques)</subfield></datafield><datafield tag="650" ind1=" " ind2="7"><subfield code="a">Sobolev ruimten</subfield><subfield code="2">gtt</subfield></datafield><datafield tag="650" ind1=" " ind2="7"><subfield code="a">Vergelijkingen (wiskunde)</subfield><subfield code="2">gtt</subfield></datafield><datafield tag="650" ind1=" " ind2="7"><subfield code="a">Desigualdades (análise matemática)</subfield><subfield code="2">larpcal</subfield></datafield><datafield tag="650" ind1=" " ind2="7"><subfield code="a">MATHEMATICS / Functional Analysis</subfield><subfield code="2">bisacsh</subfield></datafield><datafield tag="650" ind1=" " ind2="7"><subfield code="a">Inequalities (Mathematics)</subfield><subfield code="2">fast</subfield></datafield><datafield tag="650" ind1=" " ind2="7"><subfield code="a">Sobolev spaces</subfield><subfield code="2">fast</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Sobolev spaces</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Inequalities (Mathematics)</subfield></datafield><datafield tag="650" ind1="0" ind2="7"><subfield code="a">Sobolevsche Ungleichung</subfield><subfield code="0">(DE-588)4181715-1</subfield><subfield code="2">gnd</subfield><subfield code="9">rswk-swf</subfield></datafield><datafield tag="655" ind1=" " ind2="7"><subfield code="8">1\p</subfield><subfield code="0">(DE-588)4143413-4</subfield><subfield code="a">Aufsatzsammlung</subfield><subfield code="2">gnd-content</subfield></datafield><datafield tag="689" ind1="0" ind2="0"><subfield code="a">Sobolevsche Ungleichung</subfield><subfield code="0">(DE-588)4181715-1</subfield><subfield code="D">s</subfield></datafield><datafield tag="689" ind1="0" ind2=" "><subfield code="8">2\p</subfield><subfield code="5">DE-604</subfield></datafield><datafield tag="856" ind1="4" ind2="0"><subfield code="u">http://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&db=nlabk&AN=552338</subfield><subfield code="x">Aggregator</subfield><subfield code="3">Volltext</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">ZDB-4-EBA</subfield></datafield><datafield tag="883" ind1="1" ind2=" "><subfield code="8">1\p</subfield><subfield code="a">cgwrk</subfield><subfield code="d">20201028</subfield><subfield code="q">DE-101</subfield><subfield code="u">https://d-nb.info/provenance/plan#cgwrk</subfield></datafield><datafield tag="883" ind1="1" ind2=" "><subfield code="8">2\p</subfield><subfield code="a">cgwrk</subfield><subfield code="d">20201028</subfield><subfield code="q">DE-101</subfield><subfield code="u">https://d-nb.info/provenance/plan#cgwrk</subfield></datafield><datafield tag="943" ind1="1" ind2=" "><subfield code="a">oai:aleph.bib-bvb.de:BVB01-028493566</subfield></datafield><datafield tag="966" ind1="e" ind2=" "><subfield code="u">http://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&db=nlabk&AN=552338</subfield><subfield code="l">DE-1046</subfield><subfield code="p">ZDB-4-EBA</subfield><subfield code="q">FAW_PDA_EBA</subfield><subfield code="x">Aggregator</subfield><subfield code="3">Volltext</subfield></datafield><datafield tag="966" ind1="e" ind2=" "><subfield code="u">http://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&db=nlabk&AN=552338</subfield><subfield code="l">DE-1047</subfield><subfield code="p">ZDB-4-EBA</subfield><subfield code="q">FAW_PDA_EBA</subfield><subfield code="x">Aggregator</subfield><subfield code="3">Volltext</subfield></datafield></record></collection> |
| genre | 1\p (DE-588)4143413-4 Aufsatzsammlung gnd-content |
| genre_facet | Aufsatzsammlung |
| id | DE-604.BV043069374 |
| illustrated | Not Illustrated |
| indexdate | 2024-12-20T17:27:08Z |
| institution | BVB |
| isbn | 0521006074 1107360749 9780521006071 9781107360747 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-028493566 |
| oclc_num | 839304256 |
| open_access_boolean | |
| owner | DE-1046 DE-1047 |
| owner_facet | DE-1046 DE-1047 |
| physical | 1 Online-Ressource (x, 190 p.) |
| psigel | ZDB-4-EBA ZDB-4-EBA FAW_PDA_EBA |
| publishDate | 2002 |
| publishDateSearch | 2002 |
| publishDateSort | 2002 |
| publisher | Cambridge University Press |
| record_format | marc |
| series2 | London Mathematical Society lecture note series |
| spelling | Saloff-Coste, Laurent 1958- Verfasser (DE-588)115395776 aut Aspects of Sobolev-type inequalities Laurent Saloff-Coste Cambridge Cambridge University Press 2002 1 Online-Ressource (x, 190 p.) txt rdacontent c rdamedia cr rdacarrier London Mathematical Society lecture note series 289 Includes bibliographical references (p. 183-188) and index 1 - Sobolev inequalities in R[superscript n] - 7 -- - 1.1 - Sobolev inequalities - 7 -- - 1.1.2 - The proof due to Gagliardo and to Nirenberg - 9 -- - 1.1.3 - p = 1 implies p [greater than or equal] 1 - 10 -- - 1.2 - Riesz potentials - 11 -- - 1.2.1 - Another approach to Sobolev inequalities - 11 -- - 1.2.2 - Marcinkiewicz interpolation theorem - 13 -- - 1.2.3 - Proof of Sobolev Theorem 1.2.1 - 16 -- - 1.3 - Best constants - 16 -- - 1.3.1 - The case p = 1: isoperimetry - 16 -- - 1.3.2 - A complete proof with best constant for p = 1 - 18 -- - 1.3.3 - The case p > 1 - 20 -- - 1.4 - Some other Sobolev inequalities - 21 -- - 1.4.1 - The case p > n - 21 -- - 1.4.2 - The case p = n - 24 -- - 1.4.3 - Higher derivatives - 26 -- - 1.5 - Sobolev -- Poincare inequalities on balls - 29 -- - 1.5.1 - The Neumann and Dirichlet eigenvalues - 29 -- - 1.5.2 - Poincare inequalities on Euclidean balls - 30 -- - 1.5.3 - Sobolev -- Poincare inequalities - 31 -- - 2 - Moser's elliptic Harnack inequality - 33 -- - 2.1 - Elliptic operators in divergence form - 33 -- - 2.1.1 - Divergence form - 33 -- - 2.1.2 - Uniform ellipticity - 34 -- - 2.1.3 - A Sobolev-type inequality for Moser's iteration - 37 -- - 2.2 - Subsolutions and supersolutions - 38 -- - 2.2.1 - Subsolutions - 38 -- - 2.2.2 - Supersolutions - 43 -- - 2.2.3 - An abstract lemma - 47 -- - 2.3 - Harnack inequalities and continuity - 49 -- - 2.3.1 - Harnack inequalities - 49 -- - 2.3.2 - Holder continuity - 50 -- - 3 - Sobolev inequalities on manifolds - 53 -- - 3.1.1 - Notation concerning Riemannian manifolds - 53 -- - 3.1.2 - Isoperimetry - 55 -- - 3.1.3 - Sobolev inequalities and volume growth - 57 -- - 3.2 - Weak and strong Sobolev inequalities - 60 -- - 3.2.1 - Examples of weak Sobolev inequalities - 60 -- - 3.2.2 - (S[superscript [theta] subscript r,s])-inequalities: the parameters q and v - 61 -- - 3.2.3 - The case 0 < q < [infinity] - 63 -- - 3.2.4 - The case 1 = [infinity] - 66 -- - 3.2.5 - The case -[infinity] < q < 0 - 68 -- - 3.2.6 - Increasing p - 70 -- - 3.2.7 - Local versions - 72 -- - 3.3.1 - Pseudo-Poincare inequalities - 73 -- - 3.3.2 - Pseudo-Poincare technique: local version - 75 -- - 3.3.3 - Lie groups - 77 -- - 3.3.4 - Pseudo-Poincare inequalities on Lie groups - 79 -- - 3.3.5 - Ricci [greater than or equal] 0 and maximal volume growth - 82 -- - 3.3.6 - Sobolev inequality in precompact regions - 85 -- - 4 - Two applications - 87 -- - 4.1 - Ultracontractivity - 87 -- - 4.1.1 - Nash inequality implies ultracontractivity - 87 -- - 4.1.2 - The converse - 91 -- - 4.2 - Gaussian heat kernel estimates - 93 -- - 4.2.1 - The Gaffney-Davies L[superscript 2] estimate - 93 -- - 4.2.2 - Complex interpolation - 95 -- - 4.2.3 - Pointwise Gaussian upper bounds - 98 -- - 4.2.4 - On-diagonal lower bounds - 99 -- - 4.3 - The Rozenblum-Lieb-Cwikel inequality - 103 -- - 4.3.1 - The Schrodinger operator [Delta] -- V - 103 -- - 4.3.2 - The operator T[subscript V] = [Delta superscript -1]V - 105 -- - 4.3.3 - The Birman-Schwinger principle - 109 -- - 5 - Parabolic Harnack inequalities - 111 -- - 5.1 - Scale-invariant Harnack principle - 111 -- - 5.2 - Local Sobolev inequalities - 113 -- - 5.2.1 - Local Sobolev inequalities and volume growth - 113 -- - 5.2.2 - Mean value inequalities for subsolutions - 119 -- - 5.2.3 - Localized heat kernel upper bounds - 122 -- - 5.2.4 - Time-derivative upper bounds - 127 -- - 5.2.5 - Mean value inequalities for supersolutions - 128 -- - 5.3 - Poincare inequalities - 130 -- - 5.3.1 - Poincare inequality and Sobolev inequality - 131 -- - 5.3.2 - Some weighted Poincare inequalities - 133 -- - 5.3.3 - Whitney-type coverings - 135 -- - 5.3.4 - A maximal inequality and an application - 139 -- - 5.3.5 - End of the proof of Theorem 5.3.4 - 141 -- - 5.4 - Harnack inequalities and applications - 143 -- - 5.4.1 - An inequality for log u - 143 -- - 5.4.2 - Harnack inequality for positive supersolutions - 145 -- - 5.4.3 - Harnack inequalities for positive solutions - 146 -- - 5.4.4 - Holder continuity - 149 -- - 5.4.5 - Liouville theorems - 151 -- - 5.4.6 - Heat kernel lower bounds - 152 -- - 5.4.7 - Two-sided heat kernel bounds - 154 -- - 5.5 - The parabolic Harnack principle - 155 -- - 5.5.1 - Poincare, doubling, and Harnack - 157 -- - 5.5.2 - Stochastic completeness - 161 -- - 5.5.3 - Local Sobolev inequalities and the heat equation - 164 -- - 5.5.4 - Selected applications of Theorem 5.5.1 - 168 -- - 5.6.1 - Unimodular Lie groups - 172 -- - 5.6.2 - Homogeneous spaces - 175 -- - 5.6.3 - Manifolds with Ricci curvature bounded below - 176 Focusing on Poincaré, Nash and other Sobolev-type inequalities and their applications to the Laplace and heat diffusion equations on Riemannian manifolds, this text is an advanced graduate book that will also suit researchers Sobolev, Espaces de Inégalités (Mathématiques) Sobolev ruimten gtt Vergelijkingen (wiskunde) gtt Desigualdades (análise matemática) larpcal MATHEMATICS / Functional Analysis bisacsh Inequalities (Mathematics) fast Sobolev spaces fast Sobolev spaces Inequalities (Mathematics) Sobolevsche Ungleichung (DE-588)4181715-1 gnd rswk-swf 1\p (DE-588)4143413-4 Aufsatzsammlung gnd-content Sobolevsche Ungleichung (DE-588)4181715-1 s 2\p DE-604 http://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&db=nlabk&AN=552338 Aggregator Volltext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Saloff-Coste, Laurent 1958- Aspects of Sobolev-type inequalities Sobolev, Espaces de Inégalités (Mathématiques) Sobolev ruimten gtt Vergelijkingen (wiskunde) gtt Desigualdades (análise matemática) larpcal MATHEMATICS / Functional Analysis bisacsh Inequalities (Mathematics) fast Sobolev spaces fast Sobolev spaces Inequalities (Mathematics) Sobolevsche Ungleichung (DE-588)4181715-1 gnd |
| subject_GND | (DE-588)4181715-1 (DE-588)4143413-4 |
| title | Aspects of Sobolev-type inequalities |
| title_auth | Aspects of Sobolev-type inequalities |
| title_exact_search | Aspects of Sobolev-type inequalities |
| title_full | Aspects of Sobolev-type inequalities Laurent Saloff-Coste |
| title_fullStr | Aspects of Sobolev-type inequalities Laurent Saloff-Coste |
| title_full_unstemmed | Aspects of Sobolev-type inequalities Laurent Saloff-Coste |
| title_short | Aspects of Sobolev-type inequalities |
| title_sort | aspects of sobolev type inequalities |
| topic | Sobolev, Espaces de Inégalités (Mathématiques) Sobolev ruimten gtt Vergelijkingen (wiskunde) gtt Desigualdades (análise matemática) larpcal MATHEMATICS / Functional Analysis bisacsh Inequalities (Mathematics) fast Sobolev spaces fast Sobolev spaces Inequalities (Mathematics) Sobolevsche Ungleichung (DE-588)4181715-1 gnd |
| topic_facet | Sobolev, Espaces de Inégalités (Mathématiques) Sobolev ruimten Vergelijkingen (wiskunde) Desigualdades (análise matemática) MATHEMATICS / Functional Analysis Inequalities (Mathematics) Sobolev spaces Sobolevsche Ungleichung Aufsatzsammlung |
| url | http://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&db=nlabk&AN=552338 |
| work_keys_str_mv | AT saloffcostelaurent aspectsofsobolevtypeinequalities |