# Asymptotics of Robin eigenvalues on sharp infinite cones

### Konstantin Pankrashkin

Carl von Ossietzky Universität Oldenburg, Germany### Marco Vogel

Carl von Ossietzky Universität Oldenburg, Germany

## Abstract

Let $ω⊂R_{n}$ be a bounded domain with Lipschitz boundary. For $ε>0$ and $n∈N$, consider the infinite cone

and the operator $Q_{ε}$ acting as the Laplacian $u↦−Δu$ on $Ω_{ε}$ with the Robin boundary condition $∂_{ν}u=αu$ at $∂Ω_{ε}$, where $∂_{ν}$ is the outward normal derivative and $α>0$. We look at the dependence of the eigenvalues of $Q_{ε}$ on the parameter $ε$: 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 $n≥2$ and arbitrarily shaped “cross-sections” $ω$ and look at the spectral asymptotics as $ε$ 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 $j∈N$ and $α>0$, the $j$-th eigenvalue $E_{j}(Q_{ε})$ of $Q_{ε}$ exists for all sufficiently small $ε>0$ and satisfies

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

## Cite this article

Konstantin Pankrashkin, Marco Vogel, Asymptotics of Robin eigenvalues on sharp infinite cones. J. Spectr. Theory 13 (2023), no. 1, pp. 201–241

DOI 10.4171/JST/452