Optimal Shapes for the First Dirichlet Eigenvalue of the $p$-Laplacian and Dihedral symmetry

TL;DR

This paper investigates the optimization of the first Dirichlet eigenvalue of the p-Laplacian on doubly connected domains with dihedral symmetry, revealing symmetry properties and confirming a conjecture for specific cases.

Contribution

It establishes symmetry and monotonicity properties of the eigenvalue under domain rotations and confirms a conjecture for the case when n is odd and p=2.

Findings

Optimal configurations are symmetric with respect to the line through centers.

Monotonicity of eigenvalues with respect to domain rotations.

Confirmation of the conjecture for n odd and p=2.

Abstract

In this paper, we consider the optimization problem for the first Dirichlet eigenvalue $\lambda_1(\Omega)$ of the $p$-Laplacian $\Delta_p$, $1< p< \infty$, over a family of doubly connected planar domains $\Omega= B \setminus \overline{P}$, where $B$ is an open disk and $P\subsetneq B$ is a domain which is invariant under the action of a dihedral group $\mathbb{D}_n$ for some $n \geq 2,\;n\in \mathbb{N}$. We study the behaviour of $\lambda_1$ with respect to the rotations of $P$ about its center. We prove that the extremal configurations correspond to the cases where $\Omega$ is symmetric with respect to the line containing both the centers. Among these optimizing domains, the OFF configurations correspond to the minimizing ones while the ON configurations correspond to the maximizing ones. Furthermore, we obtain symmetry (periodicity) and monotonicity properties of $\lambda_1$ with respect to these rotations. In particular, we prove that the conjecture formulated in [14] for $n$ odd and $p=2$ holds true. As a consequence of our monotonicity results, we show that if the nodal set of a second eigenfunction of the $p$-Laplacian possesses a dihedral symmetry of the same order as that of $P$, then it can not enclose $P$.