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

Ariel Barton, Svitlana Mayboroda

Type: Article

Publication Date: 2016-04-12

Citations: 41

DOI: https://doi.org/10.1090/memo/1149

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 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 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. 2010 Mathematics Subject Classification. Primary 35J25, Secondary 31B20, 35C15, 46E35.

Locations

Memoirs of the American Mathematical Society - View - PDF
arXiv (Cornell University) - View - PDF

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Works Cited by This (77)

Action	Title	Year	Authors
+	An $L^p$ -estimate for the gradient of solutions of second order elliptic divergence equations	1963	Norman G. Meyers
+ PDF Chat	$H^p$ Sobolev spaces	1990	Robert S. Strichartz
+	Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems	1994	Carlos E. Kenig
+	Weighted Norm Inequalities and Related Topics	1985	José Garcı́a-Cuerva J.-L. Rubio de Francia
+	Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations	2000	Vladimir Kozlov Vladimir Maz’ya J. Roßmann
+	Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals	2002	Elias M. Stein Timothy S Murphy
+	Elliptic Problems in Nonsmooth Domains	2011	Pierre Grisvard
+	Theory of function spaces	1983	Hans Triebel
+	Function spaces on subsets of Rn	1984	Alf Jonsson Hans Wallin Jaak Peetre
+	Multiple Integrals in the Calculus of Variations	1966	Charles B. Morrey