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}:=\{(x_1,x')\in (0,\infty)\times\mathbb{R}^n\colon x'\in\varepsilon x_1\omega\}\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:=\frac{\operatorname{Vol}_{n-1} \partial\omega}{\operatorname{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)=-\frac{N_\omega^2{}\alpha^2}{(2j+n-2)^2{}\varepsilon^2}+O\Bigl(\frac{1}{\varepsilon}\Bigr)\quad \text{as $\varepsilon\to 0^+$.} The paper also covers some aspects of Sobolev spaces on infinite cones, which can be of independent interest.