On the behaviour of the first eigenvalue of the $p$-Laplacian with Robin boundary conditions as $p$ goes to $1$

Abstract

In this paper we study the $\Gamma$-limit, as $p\to 1$, of the functional $$J_p(u)=\frac{{\int }_{\Omega }|\nabla u{|}^p+\beta {\int }_{\partial \Omega }|u{|}^p}{{\int }_{\Omega }|u{|}^p},$$ where $\Omega$ is a smooth bounded open set in ${\mathbb{R}}^N$, $p>1$ and $\beta$ is a real number. Among our results, for $\beta >-1$, we derive an isoperimetric inequality for \[ Λ(Ω,β)=\inf_{u \in BV(Ω), u\not \equiv 0} \frac{\displaystyle |Du|(Ω) + \min(β,1)\int_{ \partial Ω} |u|}{\displaystyle \int_Ω|u|} \] which is the limit as $p\to 1^{+}$ of $\lambda (\Omega ,p,\beta )=\underset{u\in W^{1,p}(\Omega )}{\min }{J}_p(u).$ We show that among all bounded and smooth open sets with given volume, the ball maximizes $\Lambda (\Omega ,\beta )$ when $\beta \in$ $(-1,0)$ and minimizes $\Lambda (\Omega ,\beta )$ when $\beta \in [0,\infty )$.