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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Comparison of potential theoretic properties of rough domains


Authors: Krzysztof Burdzy and Zhen-Qing Chen
Journal: Proc. Amer. Math. Soc. 134 (2006), 3247-3253
MSC (2000): Primary 35P05; Secondary 60J45
Published electronically: June 6, 2006
MathSciNet review: 2231908
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Abstract: We discuss the relationships between the notions of intrinsic ultracontractivity, the parabolic Harnack principle, compactness of the 1-resolvent of the Neumann Laplacian, and the non-trap property for Euclidean domains with finite Lebesgue measure. In particular, we give an answer to an open problem raised by Davies and Simon in 1984 about the possible relationship between intrinsic ultracontractivity for the Dirichlet Laplacian in a domain $ D$ and compactness of the 1-resolvent of the Neumann Laplacian in $ D$.


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Additional Information

Krzysztof Burdzy
Affiliation: Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98115-4350
Email: burdzy@math.washington.edu

Zhen-Qing Chen
Affiliation: Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98115-4350
Email: zchen@math.washington.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08685-0
PII: S 0002-9939(06)08685-0
Keywords: Neumann Laplacian, trap domain, intrinsic ultracontractivity, parabolic Harnack principle.
Received by editor(s): May 17, 2005
Published electronically: June 6, 2006
Additional Notes: This research was partially supported by NSF grant DMS-0303310.
Communicated by: Michael T. Lacey
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.