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

In this paper, we study the Γ-limit, as p → 1, of the functional Jp(u) = ∫ω |∇u|p + β ∫ ∂ ω|u|p/ ∫ω|u|p, where ω is a smooth bounded open set in ℝN, p > 1 and β is a real number. Among our results, for β > - 1, we derive an isoperimetric inequality for Λ (ω, β) = infuϵBV (ω), u ≢ 0 |Du| (ω) + min (β, 1) ∫∂ ω |u|/∫ω |u| which is the limit as p → 1 + of λ λ(ω, p, β) = minuϵ W1,p (ω) Jp (u). We show that among all bounded and smooth open sets with given volume, the ball maximizes Λ (ω, β) when β ϵ (- 1, 0) and minimizes Λ (ω, β) when β ϵ [0, ∞).