Layer Potentials, the Hodge Laplacian, and Global Boundary Problems in Nonsmooth 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.

Contents

• Introduction
• Singular integrals on Lipschitz submanifolds of codimension one
• Estimates on fundamental solutions
• General second-order strongly elliptic systems
• The Dirichlet problem for the Hodge Laplacian and related operators
• Natural boundary problems for the Hodge Laplacian in Lipschitz domains
• Layer potential operators on Lipschitz domains
• Rellich type estimates for differential forms
• Fredholm properties of boundary integral operators on regular spaces
• Weak extensions of boundary derivative operators
• Localization arguments and the end of the proof of Theorem 6.2
• Harmonic fields on Lipschitz domains
• The proofs of the Theorems 5.1-5.5
• The proofs of the auxiliary lemmas
• Applications to Maxwell's equations on Lipschitz domains
• Analysis on Lipschitz manifolds
• The connection between $d_\partial$ and $d_{\partial\Omega}$
• Bibliography