Add time:08/23/2019         Source:sciencedirect.com

Let Ω⊂RN, N≥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↦−Δu in Ω with the Robin boundary condition ∂nu=αu on ∂Ω with ∂n being the outward normal derivative and α>0 being a parameter. We show that for large α the associated eigenvalues Ej(α) behave as Ej(α)∼−ϵjαν, where ν>2 and ϵj>0 depend on the dimension and the peak geometry. This is in contrast with the well-known estimate Ej(α)=O(α2) for the Lipschitz domains.

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