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" The Hodge-Laplacian : "
Dorina Mitrea, Irina Mitrea, Marius Mitrea, and Michael Taylor.
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BL
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863381
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Mitrea, Dorina,1965-
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| Title & Author
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The Hodge-Laplacian : : boundary value problems on Riemannian manifolds /\ Dorina Mitrea, Irina Mitrea, Marius Mitrea, and Michael Taylor.
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| Publication Statement
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Berlin ;Boston :: De Gruyter,, 2016.
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| Series Statement
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De Gruyter Studies in Mathematics,; Volume 64
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1 online resource
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| ISBN
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3110484382
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: 3110484390
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: 9783110484380
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: 9783110484397
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3110482665
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3110483394
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9783110482669
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9783110483390
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Includes bibliographical references.
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| Contents
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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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| Abstract
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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.
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Boundary value problems.
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Riemannian manifolds.
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Boundary value problems.
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Laplace-Operator
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MATHEMATICS-- Geometry-- General.
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Randwertproblem
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Riemannian manifolds.
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Riemannscher Raum
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| Dewey Classification
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516.373
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| LC Classification
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QA649.M58 2016
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| NLM classification
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31B1031B2531C1235A0135B2035J0835J2535J5535J5735Q6135R0142B2042B2542B3745A0545B0545E0545F1545P0547B3847G1049Q1558A1058A1258A1458A1558A3058C3558J0558J3278A30msc
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| Added Entry
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Mitrea, Irina
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Mitrea, Marius
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Taylor, Michael E.,1946-
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