Non-Lipschitz Uniform Domain Shape Optimization in Linear Acoustics
Michael Hinz, Anna Rozanova-Pierrat, Alexander Teplyaev · SIAM Journal on Control and Optimization · 2021
We introduce new parametrized classes of shape admissible domains in $\mathbb{R}^n$, $n\geq 2$, 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 $(\varepsilon,\infty)$-domains with possibly fractal boundaries that can have parts of any nonuniform Hausdorff dimension greater than 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 acoustics. A by-product of our proof is the result that the class of bounded $(\varepsilon,\infty)$-domains with fixed $\varepsilon$ is stable under Hausdorff convergence. An additional and related result is the Mosco convergence of Robin-type energy functionals on converging domains.