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

TL;DR

This paper investigates optimal shape design problems constrained by Helmholtz and spectral equations, providing existence results and first-order optimality conditions for mitigating sand transport and analyzing spectral properties.

Contribution

It establishes the existence of optimal shapes within a broad class of admissible sets and derives necessary optimality conditions for these shape optimization problems.

Findings

Existence of optimal shapes in a general admissible set.

Derivation of first-order necessary optimality conditions.

Application to sand transport mitigation and spectral problems.

Abstract

Coastal erosion describes the displacement of sand caused by the movement induced by tides, waves or currents. Some of its wave phenomena are modeled 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.