Uniqueness of diffusion on domains with rough boundaries

Abstract

Let $\Omega$ be a domain in ${\mathbf{R}}^d$ and $h(\phi )={\sum }_{k,l=1}^d({\partial }_k\phi ,c_{kl}{\partial }_l\phi )$ a quadratic form on $L_2(\Omega )$ with domain $C_c^{\infty }(\Omega )$ where the $c_{kl}$ are real symmetric $L_{\infty }(\Omega )$-functions with $C(x)=(c_{kl}(x))>0$ for almost all $x\in \Omega$. Further assume there are $a,\delta >0$ such that $a^{-1}{d}_{\Gamma }^{\delta }I\le C\le a{d}_{\Gamma }^{\delta }I$ for $d_{\Gamma }\le 1$ where $d_{\Gamma }$ is the Euclidean distance to the boundary $\Gamma$ of $\Omega$. We assume that $\Gamma$ is Ahlfors $s$-regular and if $s$, the Hausdorff dimension of $\Gamma$, is larger or equal to $d-1$ we also assume a mild uniformity property for $\Omega$ in the neighbourhood of one $z\in \Gamma$. Then we establish that $h$ is Markov unique, i.e. it has a unique Dirichlet form extension, if and only if $\delta \ge 1+(s-(d-1))$. The result applies to forms on Lipschitz domains or on a wide class of domains with $\Gamma$ 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 ${\mathbf{R}}^d$.

25 pages, 2 figures

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