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

M. Lanza de Cristoforis*, P. Musolino

*Corresponding author for this work

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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
Original languageEnglish
Pages (from-to)1-28
JournalAnnali di Matematica Pura ed Applicata
DOIs
Publication statusPublished - 24 Nov 2017

Keywords

  • periodic fundamental solution
  • elliptic differential equation
  • real analytic dependence
  • helmholtz equation
  • laplace equation

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