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Layer Potentials and Boundary-Value Problems for Second Order Elliptic Operators with Data in Besov Spaces

About this Title

Ariel Barton, Department of Mathematical Sciences, 309 SCEN, University of Arkansas, Fayetteville, AR 72701 and Svitlana Mayboroda, School of Mathematics, University of Minnesota, 127 Vincent Hall, 206 Church Street SE, Minneapolis, Minnesota 55455

Publication: Memoirs of the American Mathematical Society
Publication Year: 2016; Volume 243, Number 1149
ISBNs: 978-1-4704-1989-9 (print); 978-1-4704-3446-5 (online)
DOI: https://doi.org/10.1090/memo/1149
Published electronically: April 12, 2016
Keywords: Elliptic equation, boundary-value problem, Besov space, weighted Sobolev space
MSC: Primary 35J25; Secondary 31B20, 35C15, 46E35

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Table of Contents

Chapters

  • 1. Introduction
  • 2. Definitions
  • 3. The Main Theorems
  • 4. Interpolation, Function Spaces and Elliptic Equations
  • 5. Boundedness of Integral Operators
  • 6. Trace Theorems
  • 7. Results for Lebesgue and Sobolev Spaces: Historic Account and some Extensions
  • 8. The Green’s Formula Representation for a Solution
  • 9. Invertibility of Layer Potentials and Well-Posedness of Boundary-Value Problems
  • 10. Besov Spaces and Weighted Sobolev Spaces

Abstract

This monograph presents a comprehensive treatment of second order divergence form elliptic operators with bounded measurable $t$-independent coefficients in spaces of fractional smoothness, in Besov and weighted $L^p$ classes. We establish:

(1) Mapping properties for the double and single layer potentials, as well as the Newton potential;

(2) Extrapolation-type solvability results: the fact that solvability of the Dirichlet or Neumann boundary value problem at any given $L^p$ space automatically assures their solvability in an extended range of Besov spaces;

(3) Well-posedness for the non-homogeneous boundary value problems.

In particular, we prove well-posedness of the non-homogeneous Dirichlet problem with data in Besov spaces for operators with real, not necessarily symmetric coefficients.

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