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

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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Correspondence to Karl-Mikael Perfekt.

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Communicated by F. Lin

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Perfekt, K., Putinar, M. The Essential Spectrum of the Neumann–Poincaré Operator on a Domain with Corners. Arch Rational Mech Anal 223, 1019–1033 (2017). https://doi.org/10.1007/s00205-016-1051-6

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