Revista Matemática Iberoamericana


Full-Text PDF (403 KB) | Metadata | Table of Contents | RMI summary
Volume 35, Issue 3, 2019, pp. 925–948
DOI: 10.4171/rmi/1075

Published online: 2019-04-24

Characterization of the essential spectrum of the Neumann–Poincaré operator in 2D domains with corner via Weyl sequences

Eric Bonnetier[1] and Hai Zhang[2]

(1) Université Grenoble-Alpes, Grenoble, France
(2) The Hong Kong University of Science and Technology, Hong Kong

The Neumann–Poincaré (NP) operator naturally appears in the context of metamaterials as it may be used to represent the solutions of elliptic transmission problems via potentiel theory. In particular, its spectral properties are closely related to the well-posedness of these PDE’s, in the typical case where one considers a bounded inclusion of homogeneous plasmonic metamaterial embedded in a homogeneous background dielectric medium. In a recent work, M. Perfekt and M. Putinar have shown that the NP operator of a 2D curvilinear polygon has an essential spectrum, which depends only on the angles of the corners. Their proof is based on quasi-conformal mappings and techniques from complex-analysis. In this work, we characterise the spectrum of the NP operator for a 2D domain with corners in terms of elliptic corner singularity functions, which gives insight on the behaviour of generalized eigenmodes.

Keywords: Neumann–Poincaré operator, corner singularity, essential spectrum, Weyl sequences

Bonnetier Eric, Zhang Hai: Characterization of the essential spectrum of the Neumann–Poincaré operator in 2D domains with corner via Weyl sequences. Rev. Mat. Iberoam. 35 (2019), 925-948. doi: 10.4171/rmi/1075