**CBMS Regional Conference Series in Mathematics**

Volume: 83;
1994;
146 pp;
Softcover

MSC: Primary 35;
Secondary 42

Print ISBN: 978-0-8218-0309-7

Product Code: CBMS/83

List Price: $25.00

Individual Price: $20.00

**Electronic ISBN: 978-1-4704-2443-5
Product Code: CBMS/83.E**

List Price: $25.00

Individual Price: $20.00

# Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems

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*Carlos E. Kenig*

A co-publication of the AMS and CBMS

In recent years, there has been a great deal of activity in the study of boundary value problems with minimal smoothness assumptions on the coefficients or on the boundary of the domain in question. These problems are of interest both because of their theoretical importance and the implications for applications, and they have turned out to have profound and fascinating connections with many areas of analysis. Techniques from harmonic analysis have proved to be extremely useful in these studies, both as concrete tools in establishing theorems and as models which suggest what kind of result might be true. Kenig describes these developments and connections for the study of classical boundary value problems on Lipschitz domains and for the corresponding problems for second order elliptic equations in divergence form. He also points out many interesting problems in this area which remain open.

#### Table of Contents

# Table of Contents

## Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems

- Cover Cover11 free
- Title v6 free
- Copyright vi7 free
- Contents ix10 free
- Introduction xi12 free
- CHAPTER 1. Divergence form elliptic equations 114 free
- §1. Preliminaries 114
- §2. The classical Dirichlet problem 518
- §3. Estimates for harmonic measure 821
- §4. Fatou type theorems 1326
- §5. Area integrals, BMO (dω) and Hardy spaces 1932
- §6. The Neumann problem: variational and weak formulations 2639
- §7. The Dirichlet, Neumann and Regularity problems with L[sup(p)] data. Formulations of the problems 2841
- §8. Some general consequences of (R)[sub(qo)] and (N)[sub(qo)] 3144
- §9. Counterexamples based on the Beurling-Ahlfors theorem for quasi-conformal mappings 3851
- §10. Some approximation results 4053
- §11. Notes 4255

- CHAPTER 2. Some classes of examples and their perturbation theory 4558
- §1. The Dirichlet, Neumann and Regularity problems for the Laplacian on Lipschitz and C[sup(1)] domains 4558
- §2. The method of layer potentials for Laplace's equation on Lipschitz and C[sup(1)] domains 5063
- §3. Hardy spaces of harmonic functions on Lipschitz domains 5770
- §4. The Dirichlet, Neumann and Regularity problems for operators with radially independent coefficients 6376
- §5. The multilinear singular integral approach to the radially independent case and its perturbation theory. Extensions to the complex coefficient case and its connections with the boundedness of the Cauchy integral and Kato's square root problem 6780
- §6. Some analogies between the perturbation theory for the Dirichlet problem and classical differentiation theory 7588
- §7. Perturbation theory for the Dirichlet problem 8396
- §8. Perturbation theory for the regularity and Neumann problems 97110
- §9. Some examples related to the relationship between (R)[sub(p)], (N)[sub(p)] and (D)[sub(p)] 108121
- §10. Notes 111124

- CHAPTER 3. Epilogue: Some further results and open problems 114127
- References 134147
- Back Cover Back Cover1161

#### Readership

Advanced graduate students and researchers in the fields of harmonic analysis and elliptic partial differential equations.

#### Reviews

This book could be used in a course for advanced graduate students … it is already organized in the form of a course … will also be an excellent source book for amateurs as well as experts in this subject … The AMS, in publishing this series, has done the mathematical community a real service by providing timely and scholarly research manuscripts at such a reasonable price.

-- Bulletin of the AMS

The core of the work is devoted to the very recent state of the art.

-- Mathematical Reviews