Robin eigenvalues on domains with peaks

Let $Ω\subset\mathbb{R}^N$, $N\ge 2,$ be a bounded domain with an outward power-like peak which is assumed not too sharp in a suitable sense. We consider the Laplacian $u\mapsto -Δu$ in $Ω$ with the Robin boundary condition $\partial_n u=αu$ on $\partialΩ$ with $\partial_n$ being the outward normal derivative and $α>0$ being a parameter. We show that for large $α$ the associated eigenvalues $E_j(α)$ behave as $E_j(α)\sim -ε_j α^ν$, where $ν>2$ and $ε_j>0$ depend on the dimension and the peak geometry. This is in contrast with the well-known estimate $E_j(α)=O(α^2)$ for the Lipschitz domains.

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