On the strict monotonicity of the first eigenvalue of the $p$-Laplacian on annuli

TL;DR

This paper proves that the first eigenvalue of the $p$-Laplacian on annuli decreases strictly as the inner boundary moves outward, and explores related limit cases, also establishing nonradiality of certain eigenfunctions.

Contribution

It introduces a shape derivative approach to demonstrate strict monotonicity of the first eigenvalue with respect to the inner boundary position for the $p$-Laplacian, including limit cases.

Findings

First eigenvalue decreases as inner ball approaches outer boundary.

Limit cases as $p o 1$ and $p o ext{infinity}$ are analyzed.

Eigenfunctions on the first nontrivial Fučík spectrum curve are nonradial.

Abstract

Let $B_1$ be a ball in $\mathbb{R}^N$ centred at the origin and $B_0$ be a smaller ball compactly contained in $B_1$. For $p\in(1, \infty)$, using the shape derivative method, we show that the first eigenvalue of the $p$-Laplacian in annulus $B_1\setminus \overline{B_0}$ strictly decreases as the inner ball moves towards the boundary of the outer ball. The analogous results for the limit cases as $p \to 1$ and $p \to \infty$ are also discussed. Using our main result, further we prove the nonradiality of the eigenfunctions associated with the points on the first nontrivial curve of the Fu\v{c}ik spectrum of the $p$-Laplacian on bounded radial domains.