Analytic dependence of a periodic analog of a fundamental solution upon the periodicity parameters

Article

Abstract

We prove an analyticity result in Sobolev–Bessel potential spaces for the periodic analog of the fundamental solution of a general elliptic partial differential operator upon the parameters which determine the periodicity cell. Then we show concrete applications to the Helmholtz and the Laplace operators. In particular, we show that the periodic analogs of the fundamental solution of the Helmholtz and of the Laplace operator are jointly analytic in the spatial variable and in the parameters which determine the size of the periodicity cell. The analysis of the present paper is motivated by the application of the potential theoretic method to periodic anisotropic boundary value problems in which the “degree of anisotropy” is a parameter of the problem.

Keywords

Periodic fundamental solution Elliptic differential equation Real analytic dependence Helmholtz equation Laplace equation 

Mathematics Subject Classification

47H30 42B99 31B10 45A05 35J25 

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Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità degli Studi di PadovaPaduaItaly
  2. 2.Department of MathematicsAberystwyth UniversityCeredigionWales, UK

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