# Elliptic Partial Differential Equations with Almost-Real Coefficients

Share this page
*Ariel Barton*

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