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

• Introduction
• Definitions and the main theorem
• Useful theorems
• The Fundamental solution
• Properties of layer potentials
• Boundedness of layer potentials
• Invertibility of layer potentials and other properties
• Uniqueness of solutions
• Boundary data in \(H^1(\partial V)\)
• Concluding remarks
• Bibliography