Asymptotics of Robin eigenvalues for non-isotropic peaks

Abstract: Let $\Omega\subset \mathbb{R}^3$ be an open set such that \begin{align*} &\Omega \cap (-\delta,\delta)^3=\left{(x_1,x_2,x_3)\in \mathbb{R}^2\times(0,\delta): \, \left(\frac{x_1}{x_3^p},\frac{x_2}{x_3^q}\right)\in(-1,1)^2\right}\subset\mathbb{R}^{3}, \ &\Omega \setminus [-\delta,\delta]^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 Schr\"odinger operator which depends on $p$ and $q$.