Worst-case shape optimization for the Dirichlet energy

TL;DR

This paper investigates worst-case shape optimization for Dirichlet energy functionals, establishing existence of optimal shapes under uncertainty in the right-hand side and illustrating differences through numerical simulations.

Contribution

It introduces a framework for worst-case shape optimization with uncertain data and proves the existence of optimal shapes in this setting.

Findings

Existence of worst-case optimal shapes established.

Numerical simulations compare optimal shapes with known and uncertain data.

Worst-case optimization leads to different shapes than standard cases.

Abstract

We consider the optimization problem for a shape cost functional $F(\Omega,f)$ which depends on a domain $\Omega$ varying in a suitable admissible class and on a "right-hand side" $f$. More precisely, the cost functional $F$ is given by an integral which involves the solution $u$ of an elliptic PDE in $\Omega$ with right-hand side $f$; the boundary conditions considered are of the Dirichlet type. When the function $f$ is only known up to some degree of uncertainty, our goal is to obtain the existence of an optimal shape in the worst possible situation. Some numerical simulations are provided, showing the difference in the optimal shape between the case when $f$ is perfectly known and the case when only the worst situation is optimized.