by D. Mitrea, I. Mitrea, M. Mitrea, and M. Taylor

This monograph is devoted to a natural class of boundary problems for the Hodge Laplacian, acting on differential forms. This class includes the absolute and relative boundary problems used in the Hodge-style representation of absolute and relative cohomology classes of the underlying domain by harmonic forms.

We carry out the analysis on a class of domains in a Riemannian manifold called uniformly rectifiable domains, with particular emphasis on a subclass we call regular SKT domains, introduced by Semmes and Kenig and Toro, who called them chord-arc domains with vanishing constant.

Contents

• Chapter 1. Introduction and statement of main results
  1. First main result: absolute and relative boundary conditions
  2. Other problems involving tangential and normal components of harmonic forms
  3. Boundary value problems for Hodge-Dirac operators
  4. Dirichlet, Neumann, transmission, Poincare, and Robin-type boundary problems
  5. Structure of this monograph
• Chapter 2. Geometric concepts and tools
  1. Differential geometric preliminaries
  2. Elements of geometric measure theory
  3. Sharp integration by parts formulas for differential forms in Ahlfors regular domains
  4. Tangential and normal differential forms on Ahlfors regular sets
• Chapter 3. Harmonic layer potentials associated with the Hodge-de Rham formalism on UR domains
  1. A fundamental solution for the Hodge Laplacian
  2. Layer potentials for the Hodge-Laplacian in the Hodge-de Rham formalism
  3. Fredholm theory for layer potentials in the Hodge-de Rham formalism
• Chapter 4. Harmonic layer potentials associated with the Levi-Civita connection on UR domains
  1. The definition and mapping properties of the double layer
  2. The double layer on UR subdomains of smooth manifolds
  3. Compactness of the double layer on regular SKT domains
• Chapter 5. Dirichlet and Neumann boundary value problems for the Hodge Laplacian on regular SKT domains
  1. Functional analytic properties for harmonic layer potentials in UR domains
  2. Invertibility results for layer potentials associated with the Levi-Civita connection
  3. Solving the Dirichlet, Neumann, transmission, Poincare, and Robin boundary value problems
• Chapter 6. Fatou theorems and integral representations for the Hodge Laplacian on regular SKT domains
  1. Convergence of families of singular integral operators
  2. A Fatou theorem for the Hodge-Laplacian in regular SKT domains
  3. Spaces of harmonic fields and Green type formulas
• Chapter 7. Solvability of boundary problems for the Hodge Laplacian in the Hodge-de Rham formalism
  1. Preparatory results
  2. Solvability results
• Chapter 8. Additional results and applications
  1. de Rham cohomology on regular SKT surfaces
  2. Maxwell’s equations in regular SKT domains
  3. Dirichlet-to-Neumann operators for the Hodge-Laplacian in regular SKT domains
  4. Fatou type results with additional constraints or regularity conditions
  5. Weak tangential and normal traces in regular SKT domains with Friedrichs property
  6. The Hodge-Poisson kernel and the Hodge-harmonic measure
• Chapter 9. Further tools from differential geometry, harmonic analysis, geometric measure theory, functional analysis, paretial differential equations, and Clifford analysis
  1. Connections and covariant derivatives on vector bundles
  2. The extension of the Levi-Civita connection to differential forms
  3. The Bochner Laplacian and Weitzenbock’s formula
  4. Sobolev spaces on boundaries of Ahlfors regular domains: the Euclidean setting
  5. Sobolev spaces on boundaries of Ahlfors regular domains: the manifold setting
  6. Integrating by parts on the boundaries of Ah;fors regular domains
  7. A global Sobolev regularity result
  8. The PV harmonic layer on a UR domain
  9. Calderon-Zygmund theory on UR domains on manifolds
  10. The Fredholmness and invertibility of elliptic differential operators
  11. Compact and close-to-compact singular integral operators
  12. A sharp divergence theorem
  13. Clifford analysis rudiments
  14. Spectral theory for unbounded linear operators subject to cancellations