Asymptotics of Robin eigenvalues on sharp infinite cones

Abstract

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

and the operator $Q_{\epsilon }^{\alpha }$ acting as the Laplacian $u\mapsto -\Delta u$ on ${\Omega }_{\epsilon }$ with the Robin boundary condition ${\partial }_{\nu }u=\alpha u$ at $\partial {\Omega }_{\epsilon }$, 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

More precisely, for any fixed $j\in \mathbb{N}$ and $\alpha >0$, the $j$-th eigenvalue $E_j(Q_{\epsilon }^{\alpha })$ of $Q_{\epsilon }^{\alpha }$ exists for all sufficiently small $\epsilon >0$ and satisfies

The paper also covers some aspects of Sobolev spaces on infinite cones, which can be of independent interest.