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memo_has_moved_text();Elliptic Partial Differential Equations with Almost-Real Coefficients

Ariel Barton, School of Mathematics, University of Minnesota, Vincent Hall, 206 Church St. SE, Minneapolis, Minnesota 55455-0488

Publication: Memoirs of the American Mathematical Society
Publication Year: 2013; Volume 223, Number 1051
ISBNs: 978-0-8218-8740-0 (print); 978-0-8218-9875-8 (online)
DOI: http://dx.doi.org/10.1090/S0065-9266-2012-00677-0
Published electronically: October 24, 2012
Keywords:Elliptic equations, complex coefficients, Dirichlet problem, Neumann problem, regularity problem, Lipschitz domains, Hardy spaces, layer potentials
MSC: Primary 35J25; Secondary 31A25

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Chapters

• Chapter 1. Introduction
• Chapter 2. Definitions and the main theorem
• Chapter 3. Useful theorems
• Chapter 4. The fundamental solution
• Chapter 5. Properties of layer potentials
• Chapter 6. Boundedness of layer potentials
• Chapter 7. Invertibility of layer potentials and other properties
• Chapter 8. Uniqueness of solutions
• Chapter 9. Boundary data in Hardy spaces
• Chapter 10. Concluding remarks

Abstract

In this monograph we investigate 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.

We show that for such operators, the Dirichlet problem with boundary data in $L^q$ can be solved for $q<\infty$ large enough. We also show 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 $p=1$.