A Fourier Series Approach

  • Joachim Gwinner
  • Ernst Peter Stephan
Part of the Springer Series in Computational Mathematics book series (SSCM, volume 52)


The aim of this chapter is to guide the reader from elementary Fourier series expension to periodic Sobolev spaces on a simply connected smooth curve in \(\mathbb {R}^{2}\). In this tour we detail on dual spaces and compact embedding. This leads to the compactness of the double-layer operator and its adjoint. Moreover in the scale of Sobolev spaces we prove the mapping property of the single-layer and hypersingular operators. Then we treat the exterior Dirichlet problem for the Laplacian and derive its explicit solution on the unit circle in terms of the Fourier coefficients. The Fourier tour concludes with the first Gårding inequality for a bilinear form which is basic in the BEM.


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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Joachim Gwinner
    • 1
  • Ernst Peter Stephan
    • 2
  1. 1.Fakultät für Luft- und RaumfahrttechnikUniversität der Bundeswehr MünchenNeubiberg/MünchenGermany
  2. 2.Institut für Angewandte MathematikLeibniz Universität HannoverHannoverGermany

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