A Probabilistic Proof of the Breakdown of Besov Regularity in L-Shaped Domains
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.
KeywordsBrownian motion Dirichlet problem Poisson equation Conformal mapping Stochastic representation Besov regularity
MSC 201060J65 35C15 35J05 35J25 46E35
We thank S. Dahlke (Marburg) who pointed out the reference , 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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