  We introduce new parametrized classes of shape admissible domains in R^n , n
$\ge$ 2, and prove that they are compact with respect to the convergence in the
sense of characteristic functions, the Hausdorff sense, the sense of compacts
and the weak convergence of their boundary volumes. The domains in these
classes are bounded ($\epsilon$, $\infty$)-domains with possibly fractal
boundaries that can have parts of any non-uniform Hausdorff dimension greater
or equal to n -- 1 and less than n. We prove the existence of optimal shapes in
such classes for maximum energy dissipation in the framework of linear
acous-tics. A by-product of our proof is the result that the class of bounded
($\epsilon$, $\infty$)-domains with fixed $\epsilon$ is stable under Hausdorff
convergence. An additional and related result is the Mosco convergence of
Robin-type energy functionals on converging domains.
