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

Abstract

Abstract In this paper, we study the Γ-limit, as p → 1 {p\to 1} , of the functional J p ⁢ ( u ) = ∫ Ω | ∇ ⁡ u | p + β ⁢ ∫ ∂ ⁡ Ω | u | p ∫ Ω | u | p , J_{p}(u)=\frac{\int_{\Omega}\lvert\nabla u\rvert^{p}+\beta\int_{\partial\Omega% }\lvert u\rvert^{p}}{\int_{\Omega}\lvert u\rvert^{p}}, where Ω is a smooth bounded open set in ℝ N {\mathbb{R}^{N}} , p > 1 {p>1} and β is a real number. Among our results, for β > - 1 {\beta>-1} , we derive an isoperimetric inequality for Λ ⁢ ( Ω , β ) = inf u ∈ BV ⁡ ( Ω ) , u ≢ 0 ⁡ | D ⁢ u | ⁢ ( Ω ) + min ⁡ ( β , 1 ) ⁢ ∫ ∂ ⁡ Ω | u | ∫ Ω | u | \Lambda(\Omega,\beta)=\inf_{u\in\operatorname{BV}(\Omega),\,u\not\equiv 0}% \frac{\lvert Du\rvert(\Omega)+\min(\beta,1)\int_{\partial\Omega}\lvert u\rvert% }{\int_{\Omega}\lvert u\rvert} which is the limit as p → 1 + {p\to 1^{+}} of λ ⁢ ( Ω , p , β ) = min u ∈ W 1 , p ⁢ ( Ω ) ⁡ J p ⁢ ( u ) {\lambda(\Omega,p,\beta)=\min_{u\in W^{1,p}(\Omega)}J_{p}(u)} . We show that among all bounded and smooth open sets with given volume, the ball maximizes Λ ⁢ ( Ω , β ) {\Lambda(\Omega,\beta)} when β ∈ ( - 1 , 0 ) {\beta\in(-1,0)} and minimizes Λ ⁢ ( Ω , β ) {\Lambda(\Omega,\beta)} when β ∈ [ 0 , ∞ ) {\beta\in[0,\infty)} .