Extremal first Dirichlet eigenvalue of doubly connected plane domains and dihedral symmetry

Abstract

We deal with the following eigenvalue optimization problem: Given a bounded domain $D\subset {\mathbb{R}}^2$, how to place an obstacle $B$ of fixed shape within $D$ so as to maximize or minimize the fundamental eigenvalue ${\lambda }_1$ of 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 ${\mathbb{D}}_n$, $n\ge 2$, 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 Analysis