To find the most efficient shape of a noise-absorbing wall to dissipate the acoustical energy of a sound wave, we consider a frequency model described by the Helmholtz equation with damping on the boundary modeled by a complex-valued Robin boundary condition. We introduce a class of admissible Lipschitz boundaries, in which an optimal shape of the wall ensures the infimum of the acoustic energy. Then we also introduce a larger compact class of $(\epsilon, \infty)$ - or uniform domains with possibly non-Lipschitz (for example, fractal) boundaries on which an optimal shape exists, giving the minimum of the energy. The boundaries are described as the supports of Radon measures ensuring their Hausdorff dimension in the segment $[n−1, n)$. For a fixed Lipschitz or non-Lipschitz boundary, we also solve theoretically and numerically a parametric shape optimization problem to find the optimal distribution of absorbing material in the reflexive one minimizing the acoustical energy on a frequency range.

A by-product of the proof is that the class of bounded $(\epsilon, \infty)$-domains with fixed $\epsilon$ is stable under Hausdorff convergence. Another related result is the Mosco convergence of Robin-type energy functionals on converging domains.

Bibliography for the existence of optimal shapes

In acoustics (mixed boundary conditions Dirichlet/Neumann/Robin):

• F. Magoulès, M. Menoux , A. Rozanova-Pierrat, Frequency range non-Lipschitz parametric optimization of a noise absorption, preprint hal-04691541 (2024).

• F. Magoulès, P.T.K. Ngyuen, P. Omnes, A. Rozanova-Pierrat, Optimal absorbtion of acoustic waves by a boundary. SIAM J. Control Optim. (2021).

• M. Hinz, A. Rozanova-Pierrat, A. Teplyaev, Non-Lipschitz uniform domain shape optimization in linear acoustics. SIAM J. Control Optim. (2021).

In architecture (non-homogeneous Dirichlet and Neumann conditions):

• M. Hinz, F. Magoulès, A. Rozanova-Pierrat, M. Rynkovskaya, A. Teplyaev, On the existence of optimal shapes in architecture. Appl. Math. Model., (2021).

In heat exchanges (transmission problem):

• G. Claret, A. Rozanova-Pierrat, Existence of optimal shapes for heat diffusions across irregular interfaces, to appear in AMS, CONM book series (2024).

In the elliptic general framework with Robin type condition:

• M. Hinz, A. Rozanova-Pierrat, A. Teplyaev, Boundary value problems on Non-Lipschitz uniform domains: Stability, Compactness and the Existence of optimal shapes. Asymptotic Analysis, (2023).