eBook ISBN:  9781470403065 
Product Code:  MEMO/150/713.E 
List Price:  $57.00 
MAA Member Price:  $51.30 
AMS Member Price:  $34.20 
eBook ISBN:  9781470403065 
Product Code:  MEMO/150/713.E 
List Price:  $57.00 
MAA Member Price:  $51.30 
AMS Member Price:  $34.20 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 150; 2001; 120 ppMSC: Primary 35; 42; 58; Secondary 31; 45; 78
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 (§§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 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, §§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.
ReadershipGraduate students and research mathematicians.

Table of Contents

Chapters

Introduction

1. Singular integrals on Lipschitz submanifolds of codimension one

2. Estimates on fundamental solutions

3. General secondorder strongly elliptic systems

4. The Dirichlet problem for the Hodge Laplacian and related operators

5. Natural boundary problems for the Hodge Laplacian in Lipschitz domains

6. Layer potential operators on Lipschitz domains

7. Rellich type estimates for differential forms

8. Fredholm properties of boundary integral operators on regular spaces

9. Weak extensions of boundary derivative operators

10. Localization arguments and the end of the proof of Theorem 6.2

11. Harmonic fields on Lipschitz domains

12. The proofs of the Theorems 5.1–5.5

13. The proofs of the auxiliary lemmas

14. Applications to Maxwell’s equations on Lipschitz domains


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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 (§§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 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, §§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.
Graduate students and research mathematicians.

Chapters

Introduction

1. Singular integrals on Lipschitz submanifolds of codimension one

2. Estimates on fundamental solutions

3. General secondorder strongly elliptic systems

4. The Dirichlet problem for the Hodge Laplacian and related operators

5. Natural boundary problems for the Hodge Laplacian in Lipschitz domains

6. Layer potential operators on Lipschitz domains

7. Rellich type estimates for differential forms

8. Fredholm properties of boundary integral operators on regular spaces

9. Weak extensions of boundary derivative operators

10. Localization arguments and the end of the proof of Theorem 6.2

11. Harmonic fields on Lipschitz domains

12. The proofs of the Theorems 5.1–5.5

13. The proofs of the auxiliary lemmas

14. Applications to Maxwell’s equations on Lipschitz domains