The Hodge-Laplacian : boundary value problems on Riemannian manifolds / Dorina Mitrea, Irina Mitrea, Marius Mitrea, and Michael Taylor.
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...
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Full Text (via ProQuest) |
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| Main Authors: | , , , |
| Format: | eBook |
| Language: | English |
| Published: |
Berlin ; Boston :
De Gruyter,
2016.
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| Series: | De Gruyter studies in mathematics ;
Volume 64. |
| Subjects: |
MARC
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| 100 | 1 | |a Mitrea, Dorina, |d 1965- |e author. |1 https://id.oclc.org/worldcat/entity/E39PBJpv7q38fV6GvpykF4MpT3 | |
| 245 | 1 | 4 | |a The Hodge-Laplacian : |b boundary value problems on Riemannian manifolds / |c Dorina Mitrea, Irina Mitrea, Marius Mitrea, and Michael Taylor. |
| 264 | 1 | |a Berlin ; |a Boston : |b De Gruyter, |c 2016. | |
| 300 | |a 1 online resource | ||
| 336 | |a text |b txt |2 rdacontent | ||
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| 490 | 1 | |a De Gruyter Studies in Mathematics, |x 0179-0986 ; |v Volume 64 | |
| 588 | 0 | |a Print version record. | |
| 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. | |
| 650 | 0 | |a Boundary value problems. | |
| 650 | 7 | |a Boundary value problems |2 fast | |
| 650 | 7 | |a Riemannian manifolds |2 fast | |
| 700 | 1 | |a Mitrea, Irina, |e author. | |
| 700 | 1 | |a Mitrea, Marius, |e author. | |
| 700 | 1 | |a Taylor, Michael E., |d 1946- |e author. |1 https://id.oclc.org/worldcat/entity/E39PBJyWYxDqHd4PfggBBgYwYP | |
| 758 | |i has work: |a The Hodge-Laplacian (Text) |1 https://id.oclc.org/worldcat/entity/E39PCG49fwkG9hKCrbHBckF3Qq |4 https://id.oclc.org/worldcat/ontology/hasWork | ||
| 776 | 0 | 8 | |i Print version: |a Mitrea, Dorina. |t Hodge-Laplacian. |d Berlin, De Guyter, 2016 |z 9783110482669 |z 3110482665 |w (DLC) 2016033433 |w (OCoLC)951452997 |
| 830 | 0 | |a De Gruyter studies in mathematics ; |v Volume 64. | |
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