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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
Posted:
June 6, 2006
MathSciNet review:
2231908
Full-text PDF Free Access
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References |
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Additional Information
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  and compactness of the 1-resolvent of the Neumann Laplacian in  .
References
- [BB]
R. Bañuelos and K. Burdzy, On the ``hot spots" conjecture of J. Rauch. J. Func. Anal. 164, (1999) 1-33. MR 1694534 (2000m:35085)
- [BD]
R. Bañuelos and B. Davis, A geometrical characterization of intrinsic ultracontractivity for planar domains with boundaries given by the graphs of functions. Indiana Univ. Math. J. 41, (1992) 885-913. MR 1206335 (94g:60142)
- [BB1]
R. F. Bass and K. Burdzy, Lifetimes of conditioned diffusions. Probab. Theory Relat. Fields 91, (1992) 405-443. MR 1151804 (93e:60155)
- [BB2]
R. F. Bass and K. Burdzy, Fiber Brownian motion and the `hot spots' problem. Duke Math. J. 105, (2000) 25-58. MR 1788041 (2001g:60190)
- [BCJ]
J. Baxter, R. Chacon and N. Jain, Weak limits of stopped diffusions. Trans. Amer. Math. Soc. 293, (1986) 767-792. MR 0816325 (87e:60125)
- [BCM]
K. Burdzy, Z.-Q. Chen and D. E. Marshall, Traps for reflected Brownian motion. Math. Z. 252 (2006) 103-132.
- [C]
Z.-Q. Chen, Reflecting Brownian motions and a deletion result for Sobolev spaces of order
 . Potential Analysis, 5 (1996), 383-401. MR 1401073 (97f:46048)
- [DS1]
E. B. Davies and B. Simon, Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians. J. Funct. Anal. 59, (1984) 335-395. MR 0766493 (86e:47054)
- [DS2]
E. B. Davies and B. Simon, Spectral properties of Neumann Laplacian of horns. Geom. Funct. Anal. 2, (1992) 105-117. MR 1143665 (93g:35099)
- [D]
B. Davis, Intrinsic ultracontractivity and the Dirichlet Laplacian. J. Func. Anal. 100, (1991) 162-180. MR 1124297 (92k:35065)
- [EH]
W. D. Evans and D. J. Harris, Sobolev embeddings for generalized ridged domains. Proc. London Math. Soc. 54, (1987) 141-175. MR 0872254 (88b:46056)
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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
Posted:
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 after
28 years from publication.
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