On the first Robin eigenvalue of the Finsler $p$-Laplace operator as $p\to 1$

Abstract

Let $Ω$ be a bounded, connected, sufficiently smooth open set, $p>1$ and $β\in\mathbb R$. In this paper, we study the $Γ$-convergence, as $p\rightarrow 1^+$, of the functional \[ J_p(φ)=\frac{\int_ΩF^p(\nabla φ)dx+β\int_{\partial Ω} |φ|^pF(ν)d\mathcal{H}^{N-1}}{\int_Ω|φ|^pdx} \] where $φ\in W^{1,p}(Ω)\setminus\{0\}$ and $F$ is a sufficientely smooth norm on $\mathbb R^n$. We study the limit of the first eigenvalue $λ_1(Ω,p,β)=\inf_{\substack{φ\in W^{1,p}(Ω)\\ φ\ne 0}}J_p(φ)$, as $p\to 1^+$, that is: \begin{equation*} Λ(Ω,β)=\inf_{\substack{φ\in BV(Ω)\\ φ\not\equiv 0}}\dfrac{|Du|_F(Ω)+\min\{β,1\}\displaystyle \int_{\partial Ω}|φ|F(ν)d\mathcal H^{N-1}}{\displaystyle s\int_Ω|φ|dx}. \end{equation*} 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 $C^\infty(Ω)$ functions in the sense of strict convergence on $\mathbb R^n$ and a trace inequality in $BV$ with respect to the anisotropic total variation.