Uniqueness of diffusion on domains with rough boundaries

Abstract

Let $Ω$ be a domain in $\mathbf R^d$ and $h(φ)=\sum^d_{k,l=1}(\partial_kφ, c_{kl}\partial_lφ)$ a quadratic form on $L_2(Ω)$ with domain $C_c^\infty(Ω)$ where the $c_{kl}$ are real symmetric $L_\infty(Ω)$-functions with $C(x)=(c_{kl}(x))>0$ for almost all $x\in Ω$. Further assume there are $a, δ>0$ such that $a^{-1}d_Γ^δ\,I\le C\le a\,d_Γ^δ\,I$ for $d_Γ\le 1$ where $d_Γ$ 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 $z\inΓ$. Then we establish that $h$ is Markov unique, i.e. it has a unique Dirichlet form extension, if and only if $δ\ge 1+(s-(d-1))$. 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 $\mathbf R^2$ or the complement of a uniformly disconnected set in $\mathbf R^d$.