Calculus of Variations and Geometric Measure Theory

N. Biswas - U. Das - M. Ghosh

On the optimization of the first weighted eigenvalue

created by ghosh1 on 24 Jan 2025

[BibTeX]

Published Paper

Inserted: 24 jan 2025
Last Updated: 24 jan 2025

Journal: Proc. Roy. Soc. Edinburgh Sect. A
Year: 2022
Doi: 10.1017/prm.2022.60

ArXiv: 2109.05543 PDF

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} -\Deltap \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.