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We deal with the following eigenvalue optimization problem: Given a boundeddomain $D\subset \R^2$, how to place an obstacle $B$ of fixed shape within $D$so as to maximize or minimize the fundamental eigenvalue $\lambda_1$ of theDirichlet Laplacian on $D\setminus B$. This means that we want to extremize thefunction $\rho\mapsto \lambda_1(D\setminus \rho (B))$, where $\rho$ runs overthe set of rigid motions such that $\rho (B)\subset D$. We answer this problemin the case where both $D$ and $B$ are invariant under the action of a dihedralgroup $\mathbb{D}_n$, $n\ge2$, and where the distance from the origin to theboundary is monotonous as a function of the argument between two axes ofsymmetry. The extremal configurations correspond to the cases where the axes ofsymmetry of $B$ coincide with those of $D$.

Extremal First Dirichlet Eigenvalue of Doubly Connected Plane Domains and Dihedral Symmetry