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Archive for Rational Mechanics and Analysis

, Volume 223, Issue 2, pp 1019–1033 | Cite as

The Essential Spectrum of the Neumann–Poincaré Operator on a Domain with Corners

  • Karl-Mikael Perfekt
  • Mihai Putinar
Article

Abstract

Exploiting the homogeneous structure of a wedge in the complex plane, we compute the spectrum of the anti-linear Ahlfors–Beurling transform acting on the associated Bergman space. Consequently, the similarity equivalence between the Ahlfors–Beurling transform and the Neumann–Poincaré operator provides the spectrum of the latter integral operator on a wedge. A localization technique and conformal mapping lead to the first complete description of the essential spectrum of the Neumann–Poincaré operator on a planar domain with corners, with respect to the energy norm of the associated harmonic field.

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© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of MathematicsThe University of TennesseeKnoxvilleUSA
  2. 2.Mathematics DepartmentUniversity of CaliforniaSanta BarbaraUSA
  3. 3.School of Mathematics and StatisticsNewcastle University Newcastle upon TyneNewcastle upon TyneUK

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