We assume that Γ is Ahlfors 35855238e9d29d0f0baf65f" title="Click to view the MathML source">s-regular and if 35855238e9d29d0f0baf65f" title="Click to view the MathML source">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 or the complement of a uniformly disconnected set in .