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

ISBN-13: 9781470419899 / Angielski

Ariel Barton;Svitlana Mayboroda

Opis

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

Spis treści

• Introduction
• Definitions
• The Main theorems
• Interpolation, function spaces and elliptic equations
• Boundedness of integral operators
• Trace theorems
• Results for Lebesgue and Sobolev spaces: Historic account and some extensions
• The Green's formula representation for a solution
• Invertibility of layer potentials and well-posedness of boundary-value problems
• Besov spaces and weighted Sobolev spaces
• Bibliography.