Asymptotics of Robin eigenvalues for non-isotropic peaks

Abstract

Let $\Omega \subset {\mathbb{R}}^3$ be an open set such that \begin{align*} &Ω\cap (-δ,δ)^3=\left\{(x_1,x_2,x_3)\in \mathbb{R}^2\times(0,δ): \, \left(\frac{x_1}{x_3^p},\frac{x_2}{x_3^q}\right)\in(-1,1)^2\right\}\subset\mathbb{R}^{3}, \\ &Ω\setminus [-δ,δ]^3 \text{ is a bounded Lipschitz domain}, \end{align*} for some $\delta >0$ and $1<p<q<2$. If a set satisfies the first condition one says that it has a non-isotropic peak at $0$. Now consider the operator $Q_{\Omega }^{\alpha }$ acting as the Laplacian $u\mapsto -\Delta u$ on $\Omega$ with the Robin boundary condition ${\partial }_{\nu }u=\alpha u$ on $\partial \Omega$, where ${\partial }_{\nu }$ is the outward normal derivative. We are interested in the strong coupling asymptotics of $Q_{\Omega }^{\alpha }$. We prove that for large $\alpha$ the $j$th eigenvalue $E_j(Q_{\Omega }^{\alpha })$ behaves as $E_j(Q_{\Omega }^{\alpha })\approx {\mathcal{A}}_j{\alpha }^{\frac{2}{2-q}}$, where the constants ${\mathcal{A}}_j<0$ are eigenvalues of a one dimensional Schrödinger operator which depends on $p$ and $q$.