## 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
- First main result: absolute and relative boundary conditions
- Other problems involving tangential and normal components of harmonic forms
- Boundary value problems for Hodge-Dirac operators
- Dirichlet, Neumann, transmission, Poincare, and Robin-type boundary problems
- Structure of this monograph

- Chapter 2. Geometric concepts and tools
- Differential geometric preliminaries
- Elements of geometric measure theory
- Sharp integration by parts formulas for differential forms in Ahlfors regular domains
- Tangential and normal differential forms on Ahlfors regular sets

- Chapter 3. Harmonic layer potentials associated with the Hodge-de Rham formalism on UR domains
- A fundamental solution for the Hodge Laplacian
- Layer potentials for the Hodge-Laplacian in the Hodge-de Rham formalism
- 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
- The definition and mapping properties of the double layer
- The double layer on UR subdomains of smooth manifolds
- 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
- Functional analytic properties for harmonic layer potentials in UR domains
- Invertibility results for layer potentials associated with the Levi-Civita connection
- 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
- Convergence of families of singular integral operators
- A Fatou theorem for the Hodge-Laplacian in regular SKT domains
- Spaces of harmonic fields and Green type formulas

- Chapter 7. Solvability of boundary problems for the Hodge Laplacian in the Hodge-de Rham formalism
- Preparatory results
- Solvability results

- Chapter 8. Additional results and applications
- de Rham cohomology on regular SKT surfaces
- Maxwell’s equations in regular SKT domains
- Dirichlet-to-Neumann operators for the Hodge-Laplacian in regular SKT domains
- Fatou type results with additional constraints or regularity conditions
- Weak tangential and normal traces in regular SKT domains with Friedrichs property
- 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
- Connections and covariant derivatives on vector bundles
- The extension of the Levi-Civita connection to differential forms
- The Bochner Laplacian and Weitzenbock’s formula
- Sobolev spaces on boundaries of Ahlfors regular domains: the Euclidean setting
- Sobolev spaces on boundaries of Ahlfors regular domains: the manifold setting
- Integrating by parts on the boundaries of Ah;fors regular domains
- A global Sobolev regularity result
- The PV harmonic layer on a UR domain
- Calderon-Zygmund theory on UR domains on manifolds
- The Fredholmness and invertibility of elliptic differential operators
- Compact and close-to-compact singular integral operators
- A sharp divergence theorem
- Clifford analysis rudiments
- Spectral theory for unbounded linear operators subject to cancellations