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Coastal erosion describes the displacement of sand caused by the movementinduced by tides, waves or currents. Some of its wave phenomena are modeled byHelmholtz-type equations. Our purposes, in this paper are, first, to studyoptimal shapes obstacles to mitigate sand transport under the constraint of theHelmholtz equation. And the second side of this work is related to Dirichletand Neumann spectral problems.We show the existence of optimal shapes in ageneral admissible set of quasi open sets. And necessary optimality conditionsof first order are given in a regular framework.

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