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Regularity of the Solution of the Scalar Signorini Problem in Polygonal Domains

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Abstract

The Signorini problem for the Laplace operator is considered in a general polygonal domain. It is proved that the coincidence set consists of a finite number of boundary parts plus a finite number of isolated points. The regularity of the solution is described. In particular, we show that the leading singularity is in general \(r_i^{\pi /(2\alpha _i)}\) at transition points of Signorini to Dirichlet or Neumann conditions but \(r_i^{\pi /\alpha _i}\) at kinks of the Signorini boundary, with \(\alpha _i\) being the internal angle of the domain at these critical points.

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Acknowledgements

The authors thank Constantin Christof for pointing to an incorrect argument in a previous version of the paper. The authors thank also Christof Haubner for preparing the illustrations.

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Authors and Affiliations

  1. Institut für Mathematik und Computergestützte Simulation, Universität der Bundeswehr München, 85579, Neubiberg, Germany

    Thomas Apel

  2. Université Polytechnique Hauts-de-France, LAMAV, FR CNRS 2956, 59313, Valenciennes Cedex 9, France

    Serge Nicaise

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  1. Thomas Apel
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  2. Serge Nicaise
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Correspondence to Serge Nicaise.

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Apel, T., Nicaise, S. Regularity of the Solution of the Scalar Signorini Problem in Polygonal Domains. Results Math 75, 75 (2020). https://doi.org/10.1007/s00025-020-01202-7

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