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

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.