Electronic ISBN:  9780821898758 
Product Code:  MEMO/223/1051.E 
List Price:  $72.00 
MAA Member Price:  $64.80 
AMS Member Price:  $43.20 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 223; 2013; 106 ppMSC: Primary 35; Secondary 31;
In this monograph the author investigates divergenceform elliptic partial differential equations in twodimensional 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

Chapters

1. Introduction

2. Definitions and the Main Theorem

3. Useful Theorems

4. The Fundamental Solution

5. Properties of Layer Potentials

6. Boundedness of Layer Potentials

7. Invertibility of Layer Potentials and Other Properties

8. Uniqueness of Solutions

9. Boundary Data in \texorpdfstring{$H^1(\partial V)$}Hardy spaces

10. Concluding Remarks


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In this monograph the author investigates divergenceform elliptic partial differential equations in twodimensional 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 \(p=1\).

Chapters

1. Introduction

2. Definitions and the Main Theorem

3. Useful Theorems

4. The Fundamental Solution

5. Properties of Layer Potentials

6. Boundedness of Layer Potentials

7. Invertibility of Layer Potentials and Other Properties

8. Uniqueness of Solutions

9. Boundary Data in \texorpdfstring{$H^1(\partial V)$}Hardy spaces

10. Concluding Remarks