Spectral bounds for the Neumann-Poincaré operator on planar domains with corners

Abstract

The boundary double layer potential, or the Neumann-Poincaré operator, is studied on the Sobolev space of order 1/2 along the boundary, coinciding with the space of charges giving rise to double layer potentials with finite energy in the whole space. Poincaré’s program of studying the spectrum of the boundary double layer potential is developed in complete generality on closed Lipschitz hypersurfaces in euclidean space. Furthermore, the Neumann-Poincaré operator is realized as a singular integral transform bearing similarities to the Beurling-Ahlfors transform in 2 dimensions. As an application, in the case of planar curves with corners, bounds for the spectrum of the Neumann-Poincaré operator are derived from recent results in quasi-conformal mapping theory.

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Correspondence to Karl-Mikael Perfekt.

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The work of the first author was partially supported by The Royal Physiographic Society in Lund.

The work of the second author was partially supported by the National Science Foundation Grant DMS-10-01071.

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Perfekt, K., Putinar, M. Spectral bounds for the Neumann-Poincaré operator on planar domains with corners. JAMA 124, 39–57 (2014). https://doi.org/10.1007/s11854-014-0026-5

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Keywords

  • Angle Operator
  • Bergman Space
  • Lipschitz Domain
  • Essential Spectrum
  • Singular Integral Operator