Eigenvalue Estimates for $p$-Laplace Problems on Domains Expressed in Fermi Coordinates

TL;DR

This paper derives explicit eigenvalue bounds for Neumann p-Laplace problems on domains described via Fermi coordinates, linking geometric properties to spectral estimates and asymptotic behavior.

Contribution

It provides new sharp lower bounds for eigenvalues in domains with Fermi coordinate representations and explores their asymptotic limits as the domain shrinks.

Findings

Established lower bounds for ^{odd}(D) under geometric constraints

Identified conditions where ^{odd}(D) equals (D) for p=2

Analyzed eigenvalue asymptotics as the domain distance tends to zero

Abstract

We prove explicit and sharp eigenvalue estimates for Neumann $p$-Laplace eigenvalues in domains that admit a representation in Fermi coordinates. More precisely, if $\gamma$ denotes a non-closed curve in $\mathbb{R}^2$ symmetric with respect to the $y$-axis, let $D\subset \mathbb{R}^2$ denote the domain of points that lie on one side of $\gamma$ and within a prescribed distance $\delta(s)$ from $\gamma(s)$ (here $s$ denotes the arc length parameter for $\gamma$). Write $\mu_1^{odd}(D)$ for the lowest nonzero eigenvalue of the Neumann $p$-Laplacian with an eigenfunction that is odd with respect to the $y$-axis. For all $p>1$, we provide a lower bound on $\mu_1^{odd}(D)$ when the distance function $\delta$ and the signed curvature $k$ of $\gamma$ satisfy certain geometric constraints. In the linear case ($p=2$), we establish sufficient conditions to guarantee $\mu_1^{odd}(D)=\mu_1(D)$. We finally study the asymptotics of $\mu_1(D)$ as the distance function tends to zero. We show that in the limit, the eigenvalues converge to the lowest nonzero eigenvalue of a weighted one-dimensional Neumann $p$-Laplace problem.