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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 and Svitlana Mayboroda

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: http://dx.doi.org/10.1090/memo/1149
Published electronically: April 12, 2016
Keywords:Elliptic equation, boundary-value problem, Besov space, weighted Sobolev space

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Table of Contents


Chapters

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