# Layer Potentials, the Hodge Laplacian, and Global Boundary Problems in Nonsmooth Riemannian Manifolds

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*Dorina Mitrea; Marius Mitrea; Michael Taylor*

The general aim of the present monograph is to study
boundary-value problems for second-order elliptic
operators in Lipschitz subdomains of Riemannian manifolds.

In the first part (§§1–4), we develop a theory for Cauchy
type operators on Lipschitz submanifolds of codimension one
(focused on boundedness properties and jump relations)
and solve the \(L^p\)-Dirichlet problem, with \(p\) close to
\(2\),
for general second-order strongly elliptic systems.
The solution is represented in the form of layer potentials
and optimal nontangential maximal function estimates are established.
This analysis is carried out under smoothness assumptions (for the
coefficients of the operator, metric tensor and the underlying domain)
which are in the nature of best possible.

In the second part of the monograph, §§5–13, we further
specialize this discussion to the case of Hodge Laplacian
\(\Delta:=-d\delta-\delta d\). This time, the goal is to identify all
(pairs of) natural boundary conditions of Neumann type. Owing to the
structural richness of the higher degree case we are considering, the theory
developed here encompasses in a unitary fashion many basic PDE's of
mathematical physics. Its scope extends to also cover Maxwell's equations,
dealt with separately in §14.

The main tools are those of PDE's and harmonic analysis, occasionally
supplemented with some basic facts from algebraic topology and
differential geometry.

#### Table of Contents

# Table of Contents

## Layer Potentials, the Hodge Laplacian, and Global Boundary Problems in Nonsmooth Riemannian Manifolds

- Contents vii8 free
- Introduction 112 free
- Chapter 1. Singular integrals on Lipschitz submanifolds of codimension one 617 free
- Chapter 2. Estimates on fundamental solutions 1223
- Chapter 3. General second-order strongly elliptic systems 2738
- Chapter 4. The Dirichlet problem for the Hodge Laplacian and related operators 3950
- Chapter 5. Natural boundary problems for the Hodge Laplacian in Lipschitz domains 4556
- Chapter 6. Layer potential operators on Lipschitz domains 5465
- Chapter 7. Rellich type estimates for differential forms 6071
- Chapter 8. Fredholm properties of boundary integral operators on regular spaces 6475
- Chapter 9. Weak extensions of boundary derivative operators 7182
- Chapter 10. Localization arguments and the end of the proof of Theorem 6.2 7788
- Chapter 11. Harmonic fields on Lipschitz domains 8495
- Chapter 12. The proofs of the Theorems 5.1-5.5 91102
- Chapter 13. The proofs of the auxiliary lemmas 98109
- Chapter 14. Applications to Maxwell's equations on Lipschitz domains 104115
- Appendix A. Analysis on Lipschitz manifolds 108119
- Appendix B. The connection between d∂and d∂Ω 113124
- Bibliography 117128