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 $γ$ 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 $γ$ and within a prescribed distance $δ(s)$ from $γ(s)$ (here $s$ denotes the arc length parameter for $γ$). Write $μ_1^{odd}(D)$ 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 $μ_1^{odd}(D)$ when the distance function $δ$ and the signed curvature $k$ of $γ$ satisfy certain geometric constraints. In the linear case ($p=2$), we establish sufficient conditions to guarantee $μ_1^{odd}(D)=μ_1(D)$. We finally study the asymptotics of $μ_1(D)$ 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.

Figures (3)

• Figure 1:
• Figure 2: Left: in red the graph of $r(x)=\frac{\sin (\pi x)}{\pi x}$, in black the graph of $b(x)=(1-x)^x$, $x \in [0,1]$. Right: the graph of $b(x)-r(x)$, $x \in [0,1]$.
• Figure 3: On the left: Case $(i)$; in the middle: Case $(ii)$; on the right: Case $(iii)$

Theorems & Definitions (19)

• Theorem 1.1
• Theorem 1.2: Constant $\delta$
• Theorem 1.3: Nonconstant $\delta$
• Corollary 1.1: Constant $\delta$
• Corollary 1.2: Nonconstant $\delta$
• Theorem 1.4
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• proof