Comparison of potential theoretic properties of rough domains

Burdzy, Krzysztof; Chen, Zhen-Qing

doi:10.1090/S0002-9939-06-08685-0

Comparison of potential theoretic properties of rough domains
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by Krzysztof Burdzy and Zhen-Qing Chen PDF

Proc. Amer. Math. Soc. 134 (2006), 3247-3253 Request permission

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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