Robin eigenvalues on domains with peaks

Let $Omegasubsetmathbb{R}^N$, $Nge 2,$ be a bounded domain with an outward power-like peak which is assumed not too sharp in a suitable sense. We consider the Laplacian $umapsto -Delta u$ in $Omega$ with the Robin boundary condition $partial_n u=alpha u$ on $partialOmega$ with $partial_n$ being the outward normal derivative and $alpha>0$ being a parameter. We show that for large $alpha$ the associated eigenvalues $E_j(alpha)$ behave as $E_j(alpha)sim -epsilon_j alpha^ u$, where $ u>2$ and $epsilon_j>0$ depend on the dimension and the peak geometry. This is in contrast with the well-known estimate $E_j(alpha)=O(alpha^2)$ for the Lipschitz domains.