Robin eigenvalues on domains with peaks

Abstract

Let $\Omega \subset {\mathbb{R}}^N$, $N\ge 2,$ be a bounded domain with an outward power-like peak which is assumed not too sharp in a suitable sense. We consider the Laplacian $u\mapsto -\Delta u$ in $\Omega$ with the Robin boundary condition ${\partial }_{n}u=\alpha u$ on $\partial \Omega$ 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 $E_j(\alpha )\sim -{\epsilon }_j{\alpha }^{\nu }$, where $\nu >2$ 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.

References in corpus (2)

• Eigenvalues of Robin Laplacians in infinite sectors
• On the $p$-Laplacian with Robin boundary conditions and boundary trace theorems

Cited by in corpus (4)

• Asymptotics of Robin eigenvalues on sharp infinite cones
• Strong coupling asymptotics for $\delta$-interactions supported by curves with cusps
• Peculiar behavior of the principal Laplacian eigenvalue for large negative Robin parameters
• Laplacian eigenvalues for large negative Robin parameters on domains with outward peaks