On shape and topological optimization problems with constraints Helmholtz equation and spectral problems

Coastal erosion describes the displacement of sand caused by the movement induced by tides, waves or currents. Some of its wave phenomena are modelled by Helmholtz-type equations. Our purposes, in this paper are, first, to study optimal shapes obstacles to mitigate sand transport under the constraint of the Helmholtz equation. And the second side of this work is related to Dirichlet and Neumann spectral problems. We show the existence of optimal shapes in a general admissible set of quasi open sets. And necessary optimality conditions of first order are given in a regular framework using both shape and topological optimization. Some numerical simulations are given to represent optimal domains.