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

Source abstract

We deal with the following eigenvalue optimization problem: Given a bounded domain $D\subset {\bf R}^2$, place an obstacle B of fixed shape within D so as to maximize or minimize the fundamental eigenvalue $\lambda_1$ of the Dirichlet Laplacian on $D\setminus B$. This means that we want to extremize the function $\rho\mapsto \lambda_1(D\setminus \rho (B))$, where $\rho$ runs over the set of rigid motions such that $\rho (B)\subset D$. 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.