Estimates for the first eigenvalue of the one-dimensional $p$-Laplacian

TL;DR

This paper provides bounds and asymptotic analysis for the first eigenvalue of the one-dimensional p-Laplacian, enhancing understanding of its behavior across different p values.

Contribution

It introduces new upper and lower estimates for the first eigenvalue and explores its asymptotic behavior as p approaches 1 and infinity.

Findings

Derived bounds for the first eigenvalue of the p-Laplacian.

Analyzed asymptotic behavior as p approaches 1 and infinity.

Provided insights into the eigenvalue's dependence on p.

Abstract

In the present paper, we study the first eigenvalue $\lambda(p)$ of the one-dimensional $p$-Laplacian in the interval $(-1,1)$. We give an upper and lower estimate of $\lambda(p)$ and study its asymptotic behavior as $p \to 1+0$ or $p \to \infty$.