NON-LIPSCHITZ UNIFORM DOMAIN SHAPE OPTIMIZATION IN LINEAR ACOUSTICS

We introduce new parametrized classes of shape admissible domains in R-n, n >= 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, infinity)-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 (epsilon, infinity)-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.

Automated summary

In this paper, we introduce new parametrized classes of shape admissible domains in R-n and prove their compactness in various senses. These domains are bounded (epsilon, infinity)-domains with possibly fractal boundaries, and we demonstrate the existence of optimal shapes for maximum energy dissipation in this framework. Additionally, we establish stability and convergence results for certain classes of domains and energy functionals.