Asymptotics of Robin eigenvalues on sharp infinite cones

Let $\omega \subset {\mathbb{R}}^n$ be a bounded domain with Lipschitz boundary. For $\epsilon >0$ 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}$^{n+1}$andtheoperator$Q_{\varepsilon}^{\alpha}$actingastheLaplacian$u\mapsto-\Delta u$on$\Omega_{\varepsilon}$withtheRobinboundarycondition$\partial_\nu u=\alpha u$at$\partial\Omega_\varepsilon$,where$\partial_\nu$ is the outward normal derivative and $\alpha >0$. We look at the dependence of the eigenvalues of $Q_{\epsilon }^{\alpha }$ on the parameter $\epsilon$: 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 $n\ge 2$ and arbitrarily shaped “cross-sections” $\omega$ and look at the spectral asymptotics as $\epsilon$ 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{V}\mathrm{o}\mathrm{l}${n-1} \partial\omega }{$\mathrm{V}\mathrm{o}\mathrm{l}$n \omega}$\mathrm{.}Moreprecisely,foranyfixed$j\in $\mathbb{N}$$and$\alpha>0$the$j$theigenvalue$E_j(Q^\alpha\varepsilon)$of$Q^\alpha\varepsilon$ exists for all sufficiently small $\epsilon >0$ and satisfies $E_j(Q_{\epsilon }^{\alpha })=-{\displaystyle \frac{N_{\omega }^2{\alpha }^2}{(2j+n-2{)}^2{\epsilon }^2}}+O\left({\displaystyle \frac{1}{\epsilon }}\right)$ as $\epsilon \to 0^{+}$. The paper also covers some aspects of Sobolev spaces on infinite cones, which can be of independent interest.