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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 thesense of characteristic functions, the Hausdorff sense, the sense of compactsand the weak convergence of their boundary volumes. The domains in theseclasses are bounded ($\epsilon$, $\infty$)-domains with possibly fractalboundaries that can have parts of any non-uniform Hausdorff dimension greateror equal to n -- 1 and less than n. We prove the existence of optimal shapes insuch classes for maximum energy dissipation in the framework of linearacous-tics. A by-product of our proof is the result that the class of bounded($\epsilon$, $\infty$)-domains with fixed $\epsilon$ is stable under Hausdorffconvergence. An additional and related result is the Mosco convergence ofRobin-type energy functionals on converging domains.

Non-Lipschitz uniform domain shape optimization in linear acoustics