"Eigenvalue estimates for p-Laplace problems on domains expressed in Fe" by Jeffrey J. Langford, Barbara Brandolini et al.
 

Eigenvalue estimates for p-Laplace problems on domains expressed in Fermi coordinates

Publication Date

Winter 12-15-2024

Description

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.

Journal

Journal of Mathematical Analysis and Applications

Volume

540

Issue

2

First Page

1

Last Page

21

Department

Mathematics

DOI

https://doi.org/10.1016/j.jmaa.2024.128616

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