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 $Dsubset mathbb{R}^2$ 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.