On the optimization of the first weighted eigenvalue

TL;DR

This paper investigates the optimization of the first eigenvalue in a weighted eigenvalue problem involving the p-Laplacian, establishing existence, symmetry, and qualitative properties of extremal pairs under rearrangements.

Contribution

It proves the existence of extremal weight functions and eigenfunctions, and demonstrates their symmetry and qualitative properties for the first eigenvalue optimization problem.

Findings

Existence of minimizing and maximizing pairs of weights and eigenfunctions.

Minimizers exhibit polarization invariance and Steiner symmetry.

In annular domains, minimizers have foliated Schwarz symmetry.

Abstract

For $N\geq 2$, a bounded smooth domain $\Omega$ in $\mathbb{R}^N$, and $g_0, V_0 \in L^1_{loc}(\Omega)$, we study the optimization of the first eigenvalue for the following weighted eigenvalue problem: \begin{align*} -\Delta_p \phi + V |\phi|^{p-2}\phi = \lambda g |\phi|^{p-2}\phi \text{ in } \Omega, \quad \phi=0 \text{ on } \partial \Omega, \end{align*} where $g$ and $V$ vary over the rearrangement classes of $g_0$ and $V_0$, respectively. We prove the existence of a minimizing pair $(\underline{g},\underline{V})$ and a maximizing pair $(\overline{g},\overline{V})$ for $g_0$ and $V_0$ lying in certain Lebesgue spaces. We obtain various qualitative properties such as polarization invariance, Steiner symmetry of the minimizers as well as the associated eigenfunctions for the case $p=2$. For annular domains, we prove that the minimizers and the corresponding eigenfunctions possess the foliated Schwarz symmetry.