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Proceedings of the London Mathematical Society Advance Access originally published online on October 25, 2007
Proceedings of the London Mathematical Society 2008 96(2):273-311; doi:10.1112/plms/pdm045
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© 2007 London Mathematical Society

On the Robin problem in fractal domains

Richard F. Bass

Department of Mathematics
University of Connecticut
Storrs
CT 06269-3009
USA
bass@math.uconn.edu

Krzysztof Burdzy and Zhen-Qing Chen

Department of Mathematics
Box 354350
University of Washington
Seattle
WA 98195-4350
USA
zchen@math.washington.edu

Received 13 December 2006. Revision received 3 August 2007.

We study the solution to the Robin boundary problem for the Laplacian in a Euclidean domain. We present some families of fractal domains where the infimum of the solution to the mixed Dirichlet–Robin boundary problem is greater than 0, and some other families of domains where it is equal to 0. We also give a new result on ‘trap domains’ defined in Burdzy, Chen and Marshall (Math. Z.), that is, domains where reflecting Brownian motion takes a long time to reach the center of the domain.

2000 Mathematics Subject Classification 60H30 (primary) 35J25, 60J65 (secondary).


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