عنوان

The Hodge-Laplacian :

3110484382

3110484390

9783110484380

9783110484397

3110482665

3110483394

9783110482669

9783110483390

The Hodge-Laplacian :

[Book]

boundary value problems on Riemannian manifolds /

Dorina Mitrea, Irina Mitrea, Marius Mitrea, and Michael Taylor.

Boston :

De Gruyter,

2016.

1 online resource

De Gruyter Studies in Mathematics,

Volume 64

0179-0986 ;

Includes bibliographical references.

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.

0

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.

MIL

957926

Hodge-Laplacian.

3110482665

Boundary value problems.

Riemannian manifolds.

Boundary value problems.

Laplace-Operator

MATHEMATICS-- Geometry-- General.

Randwertproblem

Riemannian manifolds.

Riemannscher Raum

MAT-- 012000

Mitrea, Dorina,1965-

Mitrea, Irina

Mitrea, Marius

Taylor, Michael E.,1946-

20200823095642.0

pn

[Book]

Y