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 $Γ$-limit, as $p\to 1$, of the functional $$ J_{p}(u)=\frac{\displaystyle\int_Ω|\nabla u|^p + β\int_{ \partial Ω} |u|^p}{\displaystyle \int_Ω|u|^p}, $$ where $Ω$ is a smooth bounded open set in $\mathbb R^{N}$, $p>1$ and $β$ is a real number. Among our results, for $β>-1$, 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 $ λ(Ω,p,β)= \displaystyle \min_{u\in W^{1,p}(Ω)} J_{p}(u). $ We show that among all bounded and smooth open sets with given volume, the ball maximizes $Λ(Ω, β)$ when $β\in$ $(-1,0)$ and minimizes $Λ(Ω, β)$ when $β\in[0, \infty)$.