Convolution Type Operators with Symmetry in Bessel Potential Spaces

Chapter

Abstract

Convolution type operators with symmetry appear naturally in boundary value problems for elliptic PDEs in symmetric or symmetrizable domains. They are defined as truncations of translation invariant operators in a scale of Sobolev-like spaces that are convolutionally similar to subspaces of even or odd functionals. The present class, as a basic example, is closely related to the Helmholtz equation in a quadrant, where a possible solution is “symmetrically” extended to a half-plane. Explicit factorization methods allow the representation of resolvent operators in closed analytic form for a large class of boundary conditions including the two-impedance and the oblique derivative problems. Moreover they allow fine results on the regularity and asymptotic behavior of the solutions.

Keywords

Convolution type operator symmetry factorization boundary value problem quadrant diffraction explicit solution Sobolev space 

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.CIDMA and Departamento de MatemáticaUniversidade de AveiroAveiroPortugal
  2. 2.Instituto Superior TécnicoUniversidade de LisboaLisboaPortugal

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