  Given an Euclidean domain with very mild regularity properties, we prove that
there exist perturbations of the Dirichlet Laplacian of the form
$-(I+S_\epsilon)\Delta$ with $\|S_\epsilon\|_{L^2\to L^2}\leq \epsilon$ whose
high energy eigenfunctions are quantum uniquely ergodic (QUE). Moreover, if we
impose stronger regularity on the domain, the same result holds with
$\|S_\epsilon\|_{L^2\to H^\gamma}\leq \epsilon$ for $\gamma>0$ depending on the
domain. We also give a proof of a local Weyl law for domains with rough
boundaries.
