  In this article, we consider elliptic diffusion problems on random domains
with non-smooth diffusion coefficients. We start by illustrating the problems
that arise from a non-smooth diffusion coefficient by recapitulating the
corresponding regularity analysis. Then, we propose an alternative approach to
address this problem by means of a perturbation method. Based on the assumption
that the diffusion coefficient can be decomposed in a possibly deterministic,
analytic part and a rough random perturbation, we derive approximation results
in terms of the perturbations amplitude for the approximation of quantities of
interest of the solution. Numerical examples are given in order to validate and
quantify the theoretical results.
