Asymptotics of Robin eigenvalues on sharp infinite cones

Abstract

Let $\omega\subset\mathbb{R}^n$ be a bounded domain with Lipschitz boundary. For $\varepsilon>0$ and $n\in\mathbb{N}$ consider the infinite cone $\Omega_{\varepsilon}:=\big\{(x_1,x')\in (0,\infty)\times\mathbb{R}^n: x'\in\varepsilon x_1\omega\big\}\subset\mathbb{R}^{n+1}$ and the operator $Q_{\varepsilon}^{\alpha}$ acting as the Laplacian $u\mapsto-\Delta u$ on $\Omega_{\varepsilon}$ with the Robin boundary condition $\partial_\nu u=\alpha u$ at $\partial\Omega_\varepsilon$, where $\partial_\nu$ is the outward normal derivative and $\alpha>0$. We look at the dependence of the eigenvalues of $Q_\varepsilon^\alpha$ on the parameter $\varepsilon$: this problem was previously addressed for $n=1$ only (in that case, the only admissible $\omega$ are finite intervals). In the present work we consider arbitrary dimensions $n\ge2$ and arbitrarily shaped "cross-sections" $\omega$ 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 $N_\omega:=\dfrac{\mathrm{Vol}_{n-1} \partial\omega }{\mathrm{Vol}_n \omega}$. More precisely, for any fixed $j\in \mathbb{N}$ and $\alpha>0$ the $j$th eigenvalue $E_j(Q^\alpha_\varepsilon)$ of $Q^\alpha_\varepsilon$ exists for all sufficiently small $\varepsilon>0$ and satisfies $E_j(Q^\alpha_\varepsilon)=-\dfrac{N_\omega^2\,\alpha^2}{(2j+n-2)^2\,\varepsilon^2}+O\left(\dfrac{1}{\varepsilon}\right)$ as $\varepsilon\to 0^+$. The paper also covers some aspects of Sobolev spaces on infinite cones, which can be of independent interest.