Uniqueness of diffusion on domains with rough boundaries
Lehrbäck, J., & Robinson, D. W. (2016). Uniqueness of diffusion on domains with rough boundaries. Nonlinear Analysis: Theory, Methods and Applications, 131, 60-80. https://doi.org/10.1016/j.na.2015.09.007
Published inNonlinear Analysis: Theory, Methods and Applications
© 2015 Elsevier Ltd. This is an preprint version of an article whose final and definitive form has been published by Elsevier.
Let Ω be a domain in View the MathML source and View the MathML source a quadratic form on L2(Ω) with domain View the MathML source where the ckl are real symmetric L∞(Ω)-functions with C(x)=(ckl(x))>0 for almost all x∈Ω. Further assume there are a,δ>0 such that View the MathML source for dΓ≤1 where dΓ is the Euclidean distance to the boundary Γ of Ω. We assume that Γ is Ahlfors s-regular and if s, the Hausdorff dimension of Γ, is larger or equal to d−1 we also assume a mild uniformity property for Ω in the neighbourhood of one z∈Γ. Then we establish that h is Markov unique, i.e. it has a unique Dirichlet form extension, if and only if δ≥1+(s−(d−1)). The result applies to forms on Lipschitz domains or on a wide class of domains with Γ a self-similar fractal. In particular it applies to the interior or exterior of the von Koch snowflake curve in View the MathML source or the complement of a uniformly disconnected set in View the MathML source.
ISSN Search the Publication Forum0362-546X
Publication in research information system
MetadataShow full item record
Showing items with similar title or keywords.
The volume of the boundary of a Sobolev (p,q)-extension domain Koskela, Pekka; Ukhlov, Alexander; Zhu, Zheng (Elsevier, 2022)Let n≥2 and $1\leq q<p><\fz$. We prove that if Ω⊂Rn is a Sobolev (p,q)-extension domain, with additional capacitory restrictions on boundary in the case q≤n−1, n>2, then |∂Ω|=0. In the case 1≤q0.</p>
Conditional particle filters with diffuse initial distributions Karppinen, Santeri; Vihola, Matti (Springer, 2021)Conditional particle filters (CPFs) are powerful smoothing algorithms for general nonlinear/non-Gaussian hidden Markov models. However, CPFs can be inefficient or difficult to apply with diffuse initial distributions, which ...
On arithmetic sums of Ahlfors-regular sets Orponen, Tuomas (Birkhäuser, 2022)Let A,B⊂RA,B⊂R be closed Ahlfors-regular sets with dimensions dimHA=:αdimHA=:α and dimHB=:βdimHB=:β. I prove that dimH[A+θB]≥α+β⋅1−α2−αdimH[A+θB]≥α+β⋅1−α2−α for all θ∈R∖Eθ∈R∖E, where dimHE=0dimHE=0.
Ahlfors-regular distances on the Heisenberg group without biLipschitz pieces Le Donne, Enrico; Li, Sean; Rajala, Tapio (Oxford University Press; London Mathematical Society, 2017)We show that the Heisenberg group is not minimal in looking down. This answers Problem 11.15 in Fractured fractals and broken dreams by David and Semmes, or equivalently, Question 22 and hence also Question 24 in ...
Quasiconformal Jordan Domains Ikonen, Toni (Walter de Gruyter GmbH, 2021)We extend the classical Carathéodory extension theorem to quasiconformal Jordan domains (Y,dY). We say that a metric space (Y,dY) is a quasiconformal Jordan domain if the completion Y of (Y,dY) has finite Hausdor 2-measure, ...