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

  • Victoria KnopovaEmail author
  • René L. Schilling
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 229)

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.

Keywords

Brownian motion Dirichlet problem Poisson equation Conformal mapping Stochastic representation Besov regularity 

MSC 2010

60J65 35C15 35J05 35J25 46E35 

Notes

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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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut Für Mathematische StochastikFachrichtung MathematikTU Dresden, DresdenGermany

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