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

In this paper we study the $Gamma$-limit, as $p o 1$, of the functional $$ J_{p}(u)=rac{displaystyleint_Omega | abla u|^p + etaint_{ partial Omega} |u|^p}{displaystyle int_Omega |u|^p}, $$ where $Omega$ is a smooth bounded open set in $R^{N}$, $p>1$ and $eta$ is a real number. Among our results, for $eta >-1$, we derive an isoperimetric inequality for [ Lambda(Omega,eta)=inf_{u in BV(Omega), u ot equiv 0} rac{displaystyle |Du|(Omega) + min(eta,1)int_{ partial Omega} |u|}{displaystyle int_Omega |u|} ] which is the limit as $p o 1^{+}$ of $ lambda(Omega,p,eta)= ds min_{uin W^{1,p}(Omega)} J_{p}(u). $ We show that among all bounded and smooth open sets with given volume, the ball maximizes $Lambda(Omega, eta)$ when $eta in$ $(-1,0)$ and minimizes $Lambda(Omega, eta)$ when $eta in[0, infty)