Uniqueness of diffusion on domains with rough boundaries

Abstract

Let Ω be a domain inWe 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 R 2 or the complement of a uniformly disconnected set in R d .

On self-adjointness of symmetric diffusion operators

“…In principle this method of proof could be extended to establish necessary conditions for the other cases Ω = R d \{0} and Ω = R d \Π covered by Theorem 3.1. In both these cases Markov uniqueness follows if δ ≥ 2 − (d − d H ) by Theorem 1.1 of [LR16]. Therefore it would appear possible to repeat the foregoing arguments with ν δ = d…”

“…Fourthly, an alternative possible approach might be by capacity estimates. The earlier analysis of L 1 -uniqueness in [RS11] and [LR16] was based on capacity arguments with the capacity cap h defined in terms of the quadratic form h. The L 1 -analysis began with Theorems 1.2 and 1.3 of [RS11] which established that L 1 -uniqueness is equivalent both to Markov uniqueness and to the capacity condition cap h (Γ) = 0. A similar analysis of L 2 -uniqueness could then be based on the capacity cap H defined in terms of the operator H with the aim to prove that the uniqueness property is now equivalent to cap H (Γ) = 0.…”

“…This includes domains with rough, smooth, fractal or uniformly disconnected boundaries (see [Hei01] or [Sem01] for details on the Ahlfors property). In addition Proposition 3.6 of [LR16] established that (1) is a necessary condition for L 1uniqueness for almost all Ahlfors regular domains. The latter result is not quite universal because of various possible boundary pathologies such as cuspoidal points or antennae.…”

“…The sufficiency of the condition was first proved for a large variety of domains in [RS11] (see also [ERS11] Theorems 4.10 and 4.14). Subsequently, the proof was extended to all Ahlfors regular domains by Proposition 2.4 in [LR16]. This includes domains with rough, smooth, fractal or uniformly disconnected boundaries (see [Hei01] or [Sem01] for details on the Ahlfors property).…”

“…This depends not only on the order of growth, 2 − δ, but also the measure of subsets of the boundary layer. The analysis of [RS11] [ERS11] and [LR16] indicates that the condition analogous to (1) for L 2 -uniqueness should be δ ≥ 2 − (d − d H )/2 (2) and δ ≥ 2 − (d − d H )/p for L p -uniqueness. The latter condition is related to the local L p -integrability of d δ−2 Γ on the boundary layers.…”