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A Probabilistic Proof of the Breakdown of Besov Regularity in L-Shaped Domains

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Stochastic Partial Differential Equations and Related Fields (SPDERF 2016)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 229))

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Abstract

We provide a probabilistic approach in order to investigate the smoothness of the solution to the Poisson and Dirichlet problems in L-shaped domains. In particular, we obtain (probabilistic) integral representations (9), (12)–(14) for the solution. We also recover Grisvard’s classic result on the angle-dependent breakdown of the regularity of the solution measured in a Besov scale.

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Acknowledgements

We thank S. Dahlke (Marburg) who pointed out the reference [11], N. Jacob (Swansea) for his suggestions on the representation of Sobolev–Slobodetskij spaces, and A. Bendikov (Wrocław) who told us about the papers [13, 14]. We are grateful to B. Böttcher for drawing the illustrations and commenting on the first draft of this paper. Financial support from NCN grant 2014/14/M/ST1/00600 (Wrocław) for V. Knopova is gratefully acknowledged.

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Authors and Affiliations

  1. Institut Für Mathematische Stochastik, Fachrichtung Mathematik, 01062, TU Dresden, Dresden, Germany

    Victoria Knopova & René L. Schilling

Authors
  1. Victoria Knopova
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  2. René L. Schilling
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Corresponding author

Correspondence to Victoria Knopova .

Editor information

Editors and Affiliations

  1. Institute for Applied Mathematics, University of Bonn, Bonn, Germany

    Andreas Eberle

  2. Fachbereich Mathematik, Technische Universität Kaiserslautern, Kaiserslautern, Germany

    Martin Grothaus

  3. Faculty of Mathematics, Bielefeld University, Bielefeld, Germany

    Walter Hoh

  4. Faculty of Mathematics, Bielefeld University, Bielefeld, Germany

    Moritz Kassmann

  5. Institut für Mathematik, Technische Universität Berlin, Berlin, Germany

    Wilhelm Stannat

  6. Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul, Korea (Republic of)

    Gerald Trutnau

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Knopova, V., Schilling, R.L. (2018). A Probabilistic Proof of the Breakdown of Besov Regularity in L-Shaped Domains. In: Eberle, A., Grothaus, M., Hoh, W., Kassmann, M., Stannat, W., Trutnau, G. (eds) Stochastic Partial Differential Equations and Related Fields. SPDERF 2016. Springer Proceedings in Mathematics & Statistics, vol 229. Springer, Cham. https://doi.org/10.1007/978-3-319-74929-7_32

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