Asymptotics of Robin eigenvalues on sharp infinite cones

Abstract

Let $ω\subset\mathbb{R}^n$ be a bounded domain with Lipschitz boundary. For $\varepsilon>0$ and $n\in\mathbb{N}$ consider the infinite cone $Ω_{\varepsilon}:=\big\{(x_1,x')\in (0,\infty)\times\mathbb{R}^n: x'\in\varepsilon x_1ω\big\}\subset\mathbb{R}^{n+1}$ and the operator $Q_{\varepsilon}^α$ acting as the Laplacian $u\mapsto-Δu$ on $Ω_{\varepsilon}$ with the Robin boundary condition $\partial_νu=αu$ at $\partialΩ_\varepsilon$, where $\partial_ν$ is the outward normal derivative and $α>0$. We look at the dependence of the eigenvalues of $Q_\varepsilon^α$ on the parameter $\varepsilon$: this problem was previously addressed for $n=1$ only (in that case, the only admissible $ω$ are finite intervals). In the present work we consider arbitrary dimensions $n\ge2$ and arbitrarily shaped "cross-sections" $ω$ and look at the spectral asymptotics as $\varepsilon$ becomes small, i.e. as the cone becomes "sharp" and collapses to a half-line. It turns out that the main term of the asymptotics of individual eigenvalues is determined by the single geometric quantity $N_ω:=\dfrac{\mathrm{Vol}_{n-1} \partialω}{\mathrm{Vol}_n ω}$. More precisely, for any fixed $j\in \mathbb{N}$ and $α>0$ the $j$th eigenvalue $E_j(Q^α_\varepsilon)$ of $Q^α_\varepsilon$ exists for all sufficiently small $\varepsilon>0$ and satisfies $E_j(Q^α_\varepsilon)=-\dfrac{N_ω^2\,α^2}{(2j+n-2)^2\,\varepsilon^2}+O\left(\dfrac{1}{\varepsilon}\right)$ as $\varepsilon\to 0^+$. The paper also covers some aspects of Sobolev spaces on infinite cones, which can be of independent interest.