Robin eigenvalues on domains with peaks

TL;DR

This paper investigates the asymptotic behavior of Robin eigenvalues on domains with outward peaks, revealing they grow faster than quadratic in the boundary parameter, contrasting with Lipschitz domains.

Contribution

It provides the first analysis of Robin eigenvalues on domains with peaks, showing they behave as a power law with exponent greater than 2 for large boundary parameters.

Findings

Eigenvalues grow as -epsilon_j * alpha^nu with nu > 2 for large alpha.

The growth rate depends on the domain's peak geometry and dimension.

This behavior differs from the quadratic growth in Lipschitz domains.

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.