Non-Lipschitz uniform domain shape optimization in linear acoustics

TL;DR

This paper introduces new classes of possibly fractal-shaped domains in linear acoustics, proves their compactness under various convergence notions, and establishes the existence of optimal shapes for maximum energy dissipation.

Contribution

It defines and analyzes non-Lipschitz, fractal-like domain classes, proving their compactness and existence of optimal shapes in acoustic energy dissipation problems.

Findings

Domains can have fractal boundaries with non-uniform Hausdorff dimension.

Existence of optimal shapes for maximum energy dissipation is established.

Stability of domain classes under Hausdorff convergence and Mosco convergence of energy functionals.

Abstract

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.