Asymptotic behavior of the first Robin eigenvalue of nonlinear operators

Abstract

Let $\Omega$ be a bounded Lipschitz domain of ${\mathbb{R}}^N$, $N\ge 2$. In this paper, we study the asymptotic behavior of the first Robin eigenvalue of the $p$-Laplace operator as $\beta$ goes to $0$ and as $\beta$ goes to $+\infty$, deriving sharp asymptotic expansions of the eigenvalue; the expansion in the Dirichlet limit $\beta \to +\infty$ is obtained under the additional assumption that $\partial \Omega$ is of class $C^{1,1}$.