Two inequalities for the first Robin eigenvalue of the Finsler Laplacian

Let Ω ⊂ Rn, n≥ 2 , be a bounded, connected, open set with Lipschitz boundary. Let F be a suitable norm in Rn and let Δ Fu= div (Fξ(∇ u) F(∇ u)) be the so-called Finsler Laplacian, with u∈ H1(Ω). In this paper, we prove two inequalities for λF(β, Ω) , the first eigenvalue of Δ F with Robin boundary conditions involving a positive function β(x). As a consequence of our result, we obtain the asymptotic behavior of λF(β, Ω) when β is a positive constant which goes to zero