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Full Description
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.
Contents
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.
