The speaker for tomorrow's Analysis and Probability seminar is Anna Rozanova-Pierrat from CentraleSupelec, Paris-Saclay, and the title of their talk is "Existence of Optimal Shapes in Linear Acoustics in the Lipschitz and Non-Lipschitz Classes of Domains."

Abstract: 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 (ε, ∞) - or uniform domains with possibly non-Lipschitz (for example, fractal) boundaries in 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) . A byproduct of our proof is that the class of bounded (ε, ∞) -domains with fixed ε is stable under Hausdorff convergence. Another related result is the Mosco convergence of Robin-type energy functionals on converging domains.