Memoirs of the American Mathematical Society 2001; 120 pp; softcover Volume: 150 ISBN10: 082182659X ISBN13: 9780821826591 List Price: US$54 Individual Members: US$32.40 Institutional Members: US$43.20 Order Code: MEMO/150/713
 The general aim of the present monograph is to study boundaryvalue problems for secondorder elliptic operators in Lipschitz subdomains of Riemannian manifolds. In the first part (§§14), 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 secondorder 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, §§513, 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. Readership Graduate students and research mathematicians. Table of Contents  Introduction
 Singular integrals on Lipschitz submanifolds of codimension one
 Estimates on fundamental solutions
 General secondorder 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 5.15.5
 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
