The Hodge-Laplacian :: boundary value problems on Riemannian manifolds /
The core of this monograph is the development of tools to derive well-posedness results in very general geometric settings for elliptic differential operators. A new generation of Calderón-Zygmund theory is developed for variable coefficient singular integral operators, which turns out to be partic...
Gespeichert in:
| Hauptverfasser: | , , , |
|---|---|
| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Berlin ; Boston :
De Gruyter,
2016.
|
| Schriftenreihe: | De Gruyter studies in mathematics ;
Volume 64. |
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Zusammenfassung: | The core of this monograph is the development of tools to derive well-posedness results in very general geometric settings for elliptic differential operators. A new generation of Calderón-Zygmund theory is developed for variable coefficient singular integral operators, which turns out to be particularly versatile in dealing with boundary value problems for the Hodge-Laplacian on uniformly rectifiable subdomains of Riemannian manifolds via boundary layer methods. In addition to absolute and relative boundary conditions for differential forms, this monograph treats the Hodge-Laplacian equipped with classical Dirichlet, Neumann, Transmission, Poincaré, and Robin boundary conditions in regular Semmes-Kenig-Toro domains. Lying at the intersection of partial differential equations, harmonic analysis, and differential geometry, this text is suitable for a wide range of PhD students, researchers, and professionals. Contents:PrefaceIntroduction and Statement of Main ResultsGeometric Concepts and ToolsHarmonic Layer Potentials Associated with the Hodge-de Rham Formalism on UR DomainsHarmonic Layer Potentials Associated with the Levi-Civita Connection on UR DomainsDirichlet and Neumann Boundary Value Problems for the Hodge-Laplacian on Regular SKT DomainsFatou Theorems and Integral Representations for the Hodge-Laplacian on Regular SKT DomainsSolvability of Boundary Problems for the Hodge-Laplacian in the Hodge-de Rham FormalismAdditional Results and ApplicationsFurther Tools from Differential Geometry, Harmonic Analysis, Geometric Measure Theory, Functional Analysis, Partial Differential Equations, and Clifford AnalysisBibliographyIndex. |
| Beschreibung: | 1 online resource |
| Bibliographie: | Includes bibliographical references. |
| ISBN: | 3110484382 9783110484380 9783110484397 3110484390 |
| ISSN: | 0179-0986 ; |
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| 505 | 0 | |a Preface ; Contents ; 1 Introduction and Statement of Main Results ; 1.1 First Main Result: Absolute and Relative Boundary Conditions ; 1.2 Other Problems Involving Tangential and Normal Components of Harmonic Forms ; 1.3 Boundary Value Problems for Hodge-Dirac Operators; 1.4 Dirichlet, Neumann, Transmission, Poincaré, and Robin-Type Boundary Problems 1.5 Structure of the Monograph ; 2 Geometric Concepts and Tools ; 2.1 Differential Geometric Preliminaries ; 2.2 Elements of Geometric Measure Theory; 2.3 Sharp Integration by Parts Formulas for Differential Forms in Ahlfors Regular Domains 2.4 Tangential and Normal Differential Forms on Ahlfors Regular Sets ; 3 Harmonic Layer Potentials Associated with the Hodge-de Rham Formalism on UR Domains; 3.1 A Fundamental Solution for the Hodge-Laplacian 3.2 Layer Potentials for the Hodge-Laplacian in the Hodge-de Rham Formalism ; 3.3 Fredholm Theory for Layer Potentials in the Hodge-de Rham Formalism ; 4 Harmonic Layer Potentials Associated with the Levi-Civita Connection on UR Domains; 4.1 The Definition and Mapping Properties of the Double Layer 4.2 The Double Layer on UR Subdomains of Smooth Manifolds ; 4.3 Compactness of the Double Layer on Regular SKT Domains ; 5 Dirichlet and Neumann Boundary Value Problems for the Hodge-Laplacian on Regular SKT Domains. | |
| 504 | |a Includes bibliographical references. | ||
| 520 | |a The core of this monograph is the development of tools to derive well-posedness results in very general geometric settings for elliptic differential operators. A new generation of Calderón-Zygmund theory is developed for variable coefficient singular integral operators, which turns out to be particularly versatile in dealing with boundary value problems for the Hodge-Laplacian on uniformly rectifiable subdomains of Riemannian manifolds via boundary layer methods. In addition to absolute and relative boundary conditions for differential forms, this monograph treats the Hodge-Laplacian equipped with classical Dirichlet, Neumann, Transmission, Poincaré, and Robin boundary conditions in regular Semmes-Kenig-Toro domains. Lying at the intersection of partial differential equations, harmonic analysis, and differential geometry, this text is suitable for a wide range of PhD students, researchers, and professionals. Contents:PrefaceIntroduction and Statement of Main ResultsGeometric Concepts and ToolsHarmonic Layer Potentials Associated with the Hodge-de Rham Formalism on UR DomainsHarmonic Layer Potentials Associated with the Levi-Civita Connection on UR DomainsDirichlet and Neumann Boundary Value Problems for the Hodge-Laplacian on Regular SKT DomainsFatou Theorems and Integral Representations for the Hodge-Laplacian on Regular SKT DomainsSolvability of Boundary Problems for the Hodge-Laplacian in the Hodge-de Rham FormalismAdditional Results and ApplicationsFurther Tools from Differential Geometry, Harmonic Analysis, Geometric Measure Theory, Functional Analysis, Partial Differential Equations, and Clifford AnalysisBibliographyIndex. | ||
| 546 | |a In English. | ||
| 650 | 0 | |a Riemannian manifolds. |0 http://id.loc.gov/authorities/subjects/sh85114045 | |
| 650 | 0 | |a Boundary value problems. |0 http://id.loc.gov/authorities/subjects/sh85016102 | |
| 650 | 6 | |a Variétés de Riemann. | |
| 650 | 6 | |a Problèmes aux limites. | |
| 650 | 7 | |a MATHEMATICS |x Geometry |x General. |2 bisacsh | |
| 650 | 7 | |a Boundary value problems |2 fast | |
| 650 | 7 | |a Riemannian manifolds |2 fast | |
| 650 | 7 | |a Laplace-Operator |2 gnd |0 http://d-nb.info/gnd/4166772-4 | |
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| author | Mitrea, Dorina, 1965- Mitrea, Irina Mitrea, Marius Taylor, Michael E., 1946- |
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| contents | Preface ; Contents ; 1 Introduction and Statement of Main Results ; 1.1 First Main Result: Absolute and Relative Boundary Conditions ; 1.2 Other Problems Involving Tangential and Normal Components of Harmonic Forms ; 1.3 Boundary Value Problems for Hodge-Dirac Operators; 1.4 Dirichlet, Neumann, Transmission, Poincaré, and Robin-Type Boundary Problems 1.5 Structure of the Monograph ; 2 Geometric Concepts and Tools ; 2.1 Differential Geometric Preliminaries ; 2.2 Elements of Geometric Measure Theory; 2.3 Sharp Integration by Parts Formulas for Differential Forms in Ahlfors Regular Domains 2.4 Tangential and Normal Differential Forms on Ahlfors Regular Sets ; 3 Harmonic Layer Potentials Associated with the Hodge-de Rham Formalism on UR Domains; 3.1 A Fundamental Solution for the Hodge-Laplacian 3.2 Layer Potentials for the Hodge-Laplacian in the Hodge-de Rham Formalism ; 3.3 Fredholm Theory for Layer Potentials in the Hodge-de Rham Formalism ; 4 Harmonic Layer Potentials Associated with the Levi-Civita Connection on UR Domains; 4.1 The Definition and Mapping Properties of the Double Layer 4.2 The Double Layer on UR Subdomains of Smooth Manifolds ; 4.3 Compactness of the Double Layer on Regular SKT Domains ; 5 Dirichlet and Neumann Boundary Value Problems for the Hodge-Laplacian on Regular SKT Domains. |
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| discipline | Mathematik |
| format | Electronic eBook |
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| id | ZDB-4-EBA-ocn960041744 |
| illustrated | Not Illustrated |
| indexdate | 2024-11-27T13:27:26Z |
| institution | BVB |
| isbn | 3110484382 9783110484380 9783110484397 3110484390 |
| issn | 0179-0986 ; |
| language | English |
| lccn | 2016033433 |
| oclc_num | 960041744 |
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| physical | 1 online resource |
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| publishDate | 2016 |
| publishDateSearch | 2016 |
| publishDateSort | 2016 |
| publisher | De Gruyter, |
| record_format | marc |
| series | De Gruyter studies in mathematics ; |
| series2 | De Gruyter Studies in Mathematics, |
| spelling | Mitrea, Dorina, 1965- author. https://id.oclc.org/worldcat/entity/E39PBJpv7q38fV6GvpykF4MpT3 http://id.loc.gov/authorities/names/n00008814 The Hodge-Laplacian : boundary value problems on Riemannian manifolds / Dorina Mitrea, Irina Mitrea, Marius Mitrea, and Michael Taylor. Berlin ; Boston : De Gruyter, 2016. 1 online resource text txt rdacontent computer c rdamedia online resource cr rdacarrier data file rda De Gruyter Studies in Mathematics, 0179-0986 ; Volume 64 Print version record. Preface ; Contents ; 1 Introduction and Statement of Main Results ; 1.1 First Main Result: Absolute and Relative Boundary Conditions ; 1.2 Other Problems Involving Tangential and Normal Components of Harmonic Forms ; 1.3 Boundary Value Problems for Hodge-Dirac Operators; 1.4 Dirichlet, Neumann, Transmission, Poincaré, and Robin-Type Boundary Problems 1.5 Structure of the Monograph ; 2 Geometric Concepts and Tools ; 2.1 Differential Geometric Preliminaries ; 2.2 Elements of Geometric Measure Theory; 2.3 Sharp Integration by Parts Formulas for Differential Forms in Ahlfors Regular Domains 2.4 Tangential and Normal Differential Forms on Ahlfors Regular Sets ; 3 Harmonic Layer Potentials Associated with the Hodge-de Rham Formalism on UR Domains; 3.1 A Fundamental Solution for the Hodge-Laplacian 3.2 Layer Potentials for the Hodge-Laplacian in the Hodge-de Rham Formalism ; 3.3 Fredholm Theory for Layer Potentials in the Hodge-de Rham Formalism ; 4 Harmonic Layer Potentials Associated with the Levi-Civita Connection on UR Domains; 4.1 The Definition and Mapping Properties of the Double Layer 4.2 The Double Layer on UR Subdomains of Smooth Manifolds ; 4.3 Compactness of the Double Layer on Regular SKT Domains ; 5 Dirichlet and Neumann Boundary Value Problems for the Hodge-Laplacian on Regular SKT Domains. Includes bibliographical references. The core of this monograph is the development of tools to derive well-posedness results in very general geometric settings for elliptic differential operators. A new generation of Calderón-Zygmund theory is developed for variable coefficient singular integral operators, which turns out to be particularly versatile in dealing with boundary value problems for the Hodge-Laplacian on uniformly rectifiable subdomains of Riemannian manifolds via boundary layer methods. In addition to absolute and relative boundary conditions for differential forms, this monograph treats the Hodge-Laplacian equipped with classical Dirichlet, Neumann, Transmission, Poincaré, and Robin boundary conditions in regular Semmes-Kenig-Toro domains. Lying at the intersection of partial differential equations, harmonic analysis, and differential geometry, this text is suitable for a wide range of PhD students, researchers, and professionals. Contents:PrefaceIntroduction and Statement of Main ResultsGeometric Concepts and ToolsHarmonic Layer Potentials Associated with the Hodge-de Rham Formalism on UR DomainsHarmonic Layer Potentials Associated with the Levi-Civita Connection on UR DomainsDirichlet and Neumann Boundary Value Problems for the Hodge-Laplacian on Regular SKT DomainsFatou Theorems and Integral Representations for the Hodge-Laplacian on Regular SKT DomainsSolvability of Boundary Problems for the Hodge-Laplacian in the Hodge-de Rham FormalismAdditional Results and ApplicationsFurther Tools from Differential Geometry, Harmonic Analysis, Geometric Measure Theory, Functional Analysis, Partial Differential Equations, and Clifford AnalysisBibliographyIndex. In English. Riemannian manifolds. http://id.loc.gov/authorities/subjects/sh85114045 Boundary value problems. http://id.loc.gov/authorities/subjects/sh85016102 Variétés de Riemann. Problèmes aux limites. MATHEMATICS Geometry General. bisacsh Boundary value problems fast Riemannian manifolds fast Laplace-Operator gnd http://d-nb.info/gnd/4166772-4 Randwertproblem gnd http://d-nb.info/gnd/4048395-2 Riemannscher Raum gnd Mitrea, Irina, author. https://id.oclc.org/worldcat/entity/E39PCjBPBHxVXw3mppmqQYDb7d http://id.loc.gov/authorities/names/no2013034542 Mitrea, Marius, author. https://id.oclc.org/worldcat/entity/E39PCjvkx4DKybCPwqHDrQjDdP http://id.loc.gov/authorities/names/n94020722 Taylor, Michael E., 1946- author. https://id.oclc.org/worldcat/entity/E39PBJyWYxDqHd4PfggBBgYwYP http://id.loc.gov/authorities/names/n81009678 has work: The Hodge-Laplacian (Text) https://id.oclc.org/worldcat/entity/E39PCG49fwkG9hKCrbHBckF3Qq https://id.oclc.org/worldcat/ontology/hasWork Print version: Mitrea, Dorina. Hodge-Laplacian. Berlin, De Guyter, 2016 9783110482669 3110482665 (DLC) 2016033433 (OCoLC)951452997 De Gruyter studies in mathematics ; Volume 64. http://id.loc.gov/authorities/names/n83742913 FWS01 ZDB-4-EBA FWS_PDA_EBA https://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&AN=1362730 Volltext |
| spellingShingle | Mitrea, Dorina, 1965- Mitrea, Irina Mitrea, Marius Taylor, Michael E., 1946- The Hodge-Laplacian : boundary value problems on Riemannian manifolds / De Gruyter studies in mathematics ; Preface ; Contents ; 1 Introduction and Statement of Main Results ; 1.1 First Main Result: Absolute and Relative Boundary Conditions ; 1.2 Other Problems Involving Tangential and Normal Components of Harmonic Forms ; 1.3 Boundary Value Problems for Hodge-Dirac Operators; 1.4 Dirichlet, Neumann, Transmission, Poincaré, and Robin-Type Boundary Problems 1.5 Structure of the Monograph ; 2 Geometric Concepts and Tools ; 2.1 Differential Geometric Preliminaries ; 2.2 Elements of Geometric Measure Theory; 2.3 Sharp Integration by Parts Formulas for Differential Forms in Ahlfors Regular Domains 2.4 Tangential and Normal Differential Forms on Ahlfors Regular Sets ; 3 Harmonic Layer Potentials Associated with the Hodge-de Rham Formalism on UR Domains; 3.1 A Fundamental Solution for the Hodge-Laplacian 3.2 Layer Potentials for the Hodge-Laplacian in the Hodge-de Rham Formalism ; 3.3 Fredholm Theory for Layer Potentials in the Hodge-de Rham Formalism ; 4 Harmonic Layer Potentials Associated with the Levi-Civita Connection on UR Domains; 4.1 The Definition and Mapping Properties of the Double Layer 4.2 The Double Layer on UR Subdomains of Smooth Manifolds ; 4.3 Compactness of the Double Layer on Regular SKT Domains ; 5 Dirichlet and Neumann Boundary Value Problems for the Hodge-Laplacian on Regular SKT Domains. Riemannian manifolds. http://id.loc.gov/authorities/subjects/sh85114045 Boundary value problems. http://id.loc.gov/authorities/subjects/sh85016102 Variétés de Riemann. Problèmes aux limites. MATHEMATICS Geometry General. bisacsh Boundary value problems fast Riemannian manifolds fast Laplace-Operator gnd http://d-nb.info/gnd/4166772-4 Randwertproblem gnd http://d-nb.info/gnd/4048395-2 Riemannscher Raum gnd |
| subject_GND | http://id.loc.gov/authorities/subjects/sh85114045 http://id.loc.gov/authorities/subjects/sh85016102 http://d-nb.info/gnd/4166772-4 http://d-nb.info/gnd/4048395-2 |
| title | The Hodge-Laplacian : boundary value problems on Riemannian manifolds / |
| title_auth | The Hodge-Laplacian : boundary value problems on Riemannian manifolds / |
| title_exact_search | The Hodge-Laplacian : boundary value problems on Riemannian manifolds / |
| title_full | The Hodge-Laplacian : boundary value problems on Riemannian manifolds / Dorina Mitrea, Irina Mitrea, Marius Mitrea, and Michael Taylor. |
| title_fullStr | The Hodge-Laplacian : boundary value problems on Riemannian manifolds / Dorina Mitrea, Irina Mitrea, Marius Mitrea, and Michael Taylor. |
| title_full_unstemmed | The Hodge-Laplacian : boundary value problems on Riemannian manifolds / Dorina Mitrea, Irina Mitrea, Marius Mitrea, and Michael Taylor. |
| title_short | The Hodge-Laplacian : |
| title_sort | hodge laplacian boundary value problems on riemannian manifolds |
| title_sub | boundary value problems on Riemannian manifolds / |
| topic | Riemannian manifolds. http://id.loc.gov/authorities/subjects/sh85114045 Boundary value problems. http://id.loc.gov/authorities/subjects/sh85016102 Variétés de Riemann. Problèmes aux limites. MATHEMATICS Geometry General. bisacsh Boundary value problems fast Riemannian manifolds fast Laplace-Operator gnd http://d-nb.info/gnd/4166772-4 Randwertproblem gnd http://d-nb.info/gnd/4048395-2 Riemannscher Raum gnd |
| topic_facet | Riemannian manifolds. Boundary value problems. Variétés de Riemann. Problèmes aux limites. MATHEMATICS Geometry General. Boundary value problems Riemannian manifolds Laplace-Operator Randwertproblem Riemannscher Raum |
| url | https://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&AN=1362730 |
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