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

Let be a bounded, connected, sufficiently smooth open set, p ą 1 and β P R. In this paper, we study the 􀀀-convergence, as p Ñ 1`, of the functional Jppϕq “ ż Fpp∇ϕqdx ` β ż B |ϕ|p FpνqdHN´1 ż |ϕ|p dx where ϕ P W1,pp qzt0u and F is a sufficientely smooth norm on Rn. We study the limit of the first eigenvalue λ1p , p, βq “ infϕPW1,pp q ϕ‰0 Jppϕq, as p Ñ 1`, that is:  p , βq “ inf ϕPBV p q ϕı0 |Du|F p q ` mintβ, 1u ż B |ϕ| FpνqdHN´1 ż |ϕ| dx . Furthermore, for β ą ´1, we obtain an isoperimetric inequality for  p , βq depending on β. The proof uses an interior approximation result for BV p q functions by C8p q functions in the sense of strict convergence on Rn and a trace inequality in BV with respect to the anisotropic total variation.