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

TL;DR

This paper analyzes the first eigenvalue $\lambda(p)$ of the one-dimensional $p$-Laplacian on $(-1,1)$ with Dirichlet boundary, deriving an explicit formula $\lambda(p)=(p-1)\left(\dfrac{\pi}{p\sin(\pi/p)}\right)^p$ and sharp bounds that separate the regimes $p\ge2$ and $1<p<2$. It establishes analyticity of $\lambda(p)$ for all $p>1$ and derives precise asymptotics as $p\to1^+$ and $p\to\infty$, including $\lambda(p)\to1$ and $\lambda'(p)\to\infty$ at $p\to1^+$, and $\lambda(p)-p\to\dfrac{\pi^2}{6}-1$, $\lambda'(p)\to1$ as $p\to\infty$. The authors also obtain a refined expansion for $\lambda(\pi/x)-\pi/x$ and present a neat inequality for the sinc function, with implications for the behavior of nonlinear eigenvalues. The results contribute explicit, quantitative understanding of the $p$-Laplacian spectrum in one dimension and its parameter dependence.

Abstract

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

Theorems & Definitions (22)

• Theorem 1.1: KTT
• Theorem 1.2
• Lemma 1.3
• Corollary 1.4
• Theorem 1.5
• Theorem 1.6
• Corollary 1.7
• Proposition 2.1
• proof
• Lemma 2.2