The essential spectrum of the Neumann--Poincare operator on a domain with corners

TL;DR

This paper characterizes the essential spectrum of the Neumann-Poincare operator on planar domains with corners by leveraging the spectrum of related transforms and conformal mappings, advancing understanding of boundary integral operators.

Contribution

It provides the first complete description of the essential spectrum of the Neumann-Poincare operator on domains with corners using localization and conformal mapping techniques.

Findings

Spectrum of the Ahlfors-Beurling transform on wedges computed

Spectrum of the Neumann-Poincare operator on domains with corners characterized

New techniques for spectral analysis of boundary integral operators introduced

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-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.