Calculus of Variations and Geometric Measure Theory

A. M. H. Chorwadwala - M. Ghosh

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

created by ghosh1 on 24 Jan 2025

[BibTeX]

Published Paper

Inserted: 24 jan 2025
Last Updated: 24 jan 2025

Journal: J. Math. Anal. Appl.
Year: 2022
Doi: 10.1016/j.jmaa.2021.125901

ArXiv: 2108.04039 PDF

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 $P$.