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

TL;DR

This paper investigates how to optimally place a symmetric obstacle within a symmetric domain to extremize the fundamental Dirichlet eigenvalue, revealing that optimal configurations align with the symmetry axes.

Contribution

It provides a solution to the eigenvalue extremization problem for doubly connected domains with dihedral symmetry, identifying the configurations that maximize or minimize the eigenvalue.

Findings

Extremal configurations align with symmetry axes.

Optimal obstacle placement depends on dihedral symmetry.

Results apply to domains with monotonic boundary distance functions.

Abstract

We 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 $\lambda_1$ 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 $D$.