The Hodge-laplacian: Boundary Value Problems on Riemannian Manifolds (De Gruyter Studies in Mathematics) (de Gruyter Studies in Mathematics, 64)

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:**Preface Introduction and Statement of Main Results Geometric Concepts and Tools Harmonic Layer Potentials Associated with the Hodge-de Rham Formalism on UR Domains Harmonic Layer Potentials Associated with the Levi-Civita Connection on UR Domains Dirichlet and Neumann Boundary Value Problems for the Hodge-Laplacian on Regular SKT Domains Fatou Theorems and Integral Representations for the Hodge-Laplacian on Regular SKT Domains Solvability of Boundary Problems for the Hodge-Laplacian in the Hodge-de Rham Formalism Additional Results and Applications Further Tools from Differential Geometry, Harmonic Analysis, Geometric Measure Theory, Functional Analysis, Partial Differential Equations, and Clifford Analysis Bibliography Index 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 5.1 Functional Analytic Properties for Harmonic Layer Potentials in UR Domains 5.2 Invertibility Results for Layer Potentials Associated with the Levi-Civita Connection 5.3 Solving the Dirichlet, Neumann, Transmission, Poincaré, and Robin Boundary Value Problems 6 Fatou Theorems and Integral Representations for the Hodge-Laplacian on Regular SKT Domains 6.1 Convergence of Families of Singular Integral Operators 6.2 A Fatou Theorem for the Hodge-Laplacian in Regular SKT Domains 6.3 Spaces of Harmonic Fields and Green Type Formulas 7 Solvability of Boundary Problems for the Hodge-Laplacian in the Hodge-de Rham Formalism 7.1 Preparatory Results 7.2 Solvability Results 8 Additional Results and Applications 8.1 de Rham Cohomology on Regular SKT Surfaces 8.2 Maxwell’s Equations in Regular SKT Domains 8.3 Dirichlet-to-Neumann Operators for the Hodge-Laplacian in Regular SKT Domains 8.4 Fatou Type Results with Additional Constraints or Regularity Conditions 8.5 Weak Tangential and Normal Traces in Regular SKT Domains with Friedrichs Property 8.6 The Hodge-Poisson Kernel and the Hodge-Harmonic Measure 9 Further Tools from Differential Geometry, Harmonic Analysis, Geometric Measure Theory, Functional Analysis, Partial Differential Equations, and Clifford Analysis 9.1 Connections and Covariant Derivatives on Vector Bundles 9.2 The Extension of the Levi-Civita Connection to Differential Forms 9.3 The Bochner-Laplacian and Weintzenböck’s Formula 9.4 Sobolev Spaces on Boundaries of Ahlfors Regular Domains: The Euclidean Setting 9.5 Sobolev Spaces on Boundaries of Ahlfors Regular Domains: The Manifold Setting 9.6 Integrating by Parts on the Boundaries of Ahlfors Regular Domains 9.7 A Global Sobolev Regularity Result 9.8 The PV Harmonic Double Layer on a UR Domain 9.9 Calderón-Zygmund Theory on UR Domains on Manifolds 9.10 The Fredholmness and Invertibility of Elliptic Differential Operators 9.11 Compact and Close-to-Compact Singular Integral Operators 9.12 A Sharp Divergence Theorem 9.13 Clifford Analysis Rudiments 9.14 Spectral Theory for Unbounded Linear Operators Subject to Cancellations Bibliography Index

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:
Preface
Introduction and Statement of Main Results
Geometric Concepts and Tools
Harmonic Layer Potentials Associated with the Hodge-de Rham Formalism on UR Domains
Harmonic Layer Potentials Associated with the Levi-Civita Connection on UR Domains
Dirichlet and Neumann Boundary Value Problems for the Hodge-Laplacian on Regular SKT Domains
Fatou Theorems and Integral Representations for the Hodge-Laplacian on Regular SKT Domains
Solvability of Boundary Problems for the Hodge-Laplacian in the Hodge-de Rham Formalism
Additional Results and Applications
Further Tools from Differential Geometry, Harmonic Analysis, Geometric Measure Theory, Functional Analysis, Partial Differential Equations, and Clifford Analysis
Bibliography
Index

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: Preface Introduction and Statement of Main Results Geometric Concepts and Tools Harmonic Layer Potentials Associated with the Hodge-de Rham Formalism on UR Domains Harmonic Layer Potentials Associated with the Levi-Civita Connection on UR Domains Dirichlet and Neumann Boundary Value Problems for the Hodge-Laplacian on Regular SKT Domains Fatou Theorems and Integral Representations for the Hodge-Laplacian on Regular SKT Domains Solvability of Boundary Problems for the Hodge-Laplacian in the Hodge-de Rham Formalism Additional Results and Applications Further Tools from Differential Geometry, Harmonic Analysis, Geometric Measure Theory, Functional Analysis, Partial Differential Equations, and Clifford Analysis Bibliography Index 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 Calderon-Zygmund theory is developed for variable coefficient singular integral operators.