**Memoirs of the American Mathematical Society**

2013;
106 pp;
Softcover

MSC: Primary 35;
Secondary 31

Print ISBN: 978-0-8218-8740-0

Product Code: MEMO/223/1051

List Price: $72.00

AMS Member Price: $43.20

MAA Member Price: $64.80

**Electronic ISBN: 978-0-8218-9875-8
Product Code: MEMO/223/1051.E**

List Price: $72.00

AMS Member Price: $43.20

MAA Member Price: $64.80

# Elliptic Partial Differential Equations with Almost-Real Coefficients

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*Ariel Barton*

In this monograph the author investigates divergence-form elliptic
partial differential equations in two-dimensional Lipschitz domains whose
coefficient matrices have small (but possibly nonzero) imaginary parts and
depend only on one of the two coordinates.

He shows that for such operators, the Dirichlet problem with boundary data
in \(L^q\) can be solved for \(q<\infty\) large enough. He also
shows that the Neumann and regularity problems with boundary data in
\(L^p\) can be solved for \(p>1\) small enough, and provide an
endpoint result at \(p=1\).

#### Table of Contents

# Table of Contents

## Elliptic Partial Differential Equations with Almost-Real Coefficients

- Chapter 1. Introduction 18 free
- 1.1. History 310 free

- Chapter 2. Definitions and the Main Theorem 916
- Chapter 3. Useful Theorems 2128
- Chapter 4. The Fundamental Solution 3340
- Chapter 5. Properties of Layer Potentials 4350
- Chapter 6. Boundedness of Layer Potentials 4956
- 6.1. Proof for a small Lipschitz constant: preliminary remarks 4956
- 6.2. A B for the TB theorem 5158
- 6.3. Weak boundedness of operators 5461
- 6.4. The adjoint inequalities 5663
- 6.5. Proof for a small Lipschitz constant: final remarks 6370
- 6.6. Buildup to arbitrary special Lipschitz domains 6471
- 6.7. Patching: special Lipschitz domains to bounded Lipschitz domains 6774

- Chapter 7. Invertibility of Layer Potentials and Other Properties 6976
- Chapter 8. Uniqueness of Solutions 8390
- Chapter 9. Boundary Data in Hardy spaces 8996
- Chapter 10. Concluding Remarks 97104
- Bibliography 105112