2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/143704We deal with the following eigenvalue optimization problem: Given a bounded domain $D\subset \R^2$, how to place an obstacle $B$ of fixed shape within $D$ so as to maximize or minimize the fundamental eigenvalue $λ_1$ of the Dirichlet Laplacian on $D\setminus B$. This means that we want to extremize the function $ρ\mapsto λ_1(D\setminus ρ(B))$, where $ρ$ runs over the set of rigid motions such that $ρ(B)\subset D$. We answer this problem in the case where both $D$ and $B$ are invariant under the action of a dihedral group $\mathbb{D}_n$, $n\ge2$, and where the distance from the origin to the boundary is monotonous as a function of the argument between two axes of symmetry. The extremal configurations correspond to the cases where the axes of symmetry of $B$ coincide with those of $D$.To appear in SIAM Journal on Mathematical AnalysisSpectral TheoryOptimization and Control35J10, 35P15, 49R50, 58J50text