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

Let $\mathrm{\Omega }$ be a bounded, connected, sufficiently smooth open set, $p>1$ and $\beta \in \mathbb{R}$. In this paper, we study the $\mathrm{\Gamma }$-convergence, as $p\to 1^{+}$, of the functional $$J_p(\phi )=\frac{{\int }_{\mathrm{\Omega }}{F}^p(\mathrm{\nabla }\phi )dx+\beta {\int }_{\mathrm{\partial }\mathrm{\Omega }}\mathrm{\mid }\phi {\mathrm{\mid }}^{p}F(\nu )d{\mathcal{H}}^{N-1}}{{\int }_{\mathrm{\Omega }}\mathrm{\mid }\phi {\mathrm{\mid }}^{p}dx}$$ where $\phi \in W^{1,p}(\mathrm{\Omega })\setminus \{0\}$ and $F$ is a sufficientely smooth norm on ${\mathbb{R}}^n$. We study the limit of the first eigenvalue ${\lambda }_1(\mathrm{\Omega },p,\beta )={\inf }_{\begin{array}{c}\phi \in W^{1,p}(\mathrm{\Omega })\\ \phi \mathrm{\ne }0\end{array}}{J}_p(\phi )$, as $p\to 1^{+}$, that is: \begin{equation*} Λ(Ω,β)=\inf_{\substack{\varphi \in BV(Ω)\ \varphi\not\equiv 0}}\dfrac{|Du|F(Ω)+\min{β,1}\displaystyle \int{\partial Ω}|\varphi|F(ν)d\mathcal H^{N-1}}{\displaystyle s\int_Ω|\varphi|dx}. \end{equation*} Furthermore, for $\beta >-1$, we obtain an isoperimetric inequality for $\mathrm{\Lambda }(\mathrm{\Omega },\beta )$ depending on $\beta$. The proof uses an interior approximation result for $BV(\mathrm{\Omega })$ functions by $C^{\mathrm{\infty }}(\mathrm{\Omega })$ functions in the sense of strict convergence on ${\mathbb{R}}^n$ and a trace inequality in $BV$ with respect to the anisotropic total variation.