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.