The essential spectrum of the Neumann–Poincaré operator on a domain with corners

Perfekt, K.-M. and Putinar, M. (2017) The essential spectrum of the Neumann–Poincaré operator on a domain with corners. Archive for Rational Mechanics and Analysis, 223 (2). pp. 1019-1033. ISSN 0003-9527 doi: 10.1007/s00205-016-1051-6

Abstract/Summary

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-Poincare 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-Poincare operator on a planar domain with corners, with respect to the energy norm of the associated harmonic field.