Asymptotics of Robin eigenvalues for non-isotropic peaks

Let $\mathrm{\Omega }\subset {\mathbb{R}}^3$ 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 $Q_{\mathrm{\Omega }}^{\alpha }$ acting as the Laplacian $u\mapsto -\mathrm{\Delta }u$ on $\mathrm{\Omega }$ with the Robin boundary condition ${\mathrm{\partial }}_{\nu }u=\alpha u$ on $\mathrm{\partial }\mathrm{\Omega }$, where ${\mathrm{\partial }}_{\nu }$ is the outward normal derivative. We are interested in the strong coupling asymptotics of $Q_{\mathrm{\Omega }}^{\alpha }$. We prove that for large $\alpha$ the $j$th eigenvalue $E_j(Q_{\mathrm{\Omega }}^{\alpha })$ behaves as $E_j(Q_{\mathrm{\Omega }}^{\alpha })\approx {\mathcal{A}}_j{\alpha }^{\frac{2}{2-q}}$, where the constants ${\mathcal{A}}_j<0$ are eigenvalues of a one dimensional Schrödinger operator which depends on $p$ and $q$.