On the first Robin eigenvalue of the Finsler p-Laplace operator as p → 1

Let Ω be a bounded, connected, sufficiently smooth open set, p>1 and β∈R. In this paper, we study the Γ-convergence, as p→1+, of the functionalJp(φ)=∫ΩFp(∇φ)dx+β∫∂Ω|φ|pF(ν)dHN−1∫Ω|φ|pdx where φ∈W1,p(Ω)∖{0} and F is a sufficiently smooth norm on Rn. We study the limit of the first eigenvalue λ1(Ω,p,β)=infφ∈W1,p(Ω)φ≠0⁡Jp(φ), as p→1+, that is:Λ(Ω,β)=infφ∈BV(Ω)φ≢0⁡|Du|F(Ω)+min⁡{β,1}∫∂Ω|φ|F(ν)dHN−1∫Ω|φ|dx. Furthermore, for β>−1, we obtain an isoperimetric inequality for Λ(Ω,β) depending on β. The proof uses an interior approximation result for BV(Ω) functions by C∞(Ω) functions in the sense of strict convergence on Rn and a trace inequality in BV with respect to the anisotropic total variation.