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Let $\Omega$ be a bounded, connected, sufficiently smooth open set, $p>1$ and$\beta\in\mathbb R$. In this paper, we study the $\Gamma$-convergence, as$p\rightarrow 1^+$, of the functional \[ J_p(\varphi)=\frac{\int_\OmegaF^p(\nabla \varphi)dx+\beta\int_{\partial \Omega}|\varphi|^pF(\nu)d\mathcal{H}^{N-1}}{\int_\Omega |\varphi|^pdx} \]  where $\varphi\in W^{1,p}(\Omega)\setminus\{0\}$ and $F$ is a sufficientelysmooth norm on $\mathbb R^n$. We study the limit of the first eigenvalue$\lambda_1(\Omega,p,\beta)=\inf_{\substack{\varphi\in W^{1,p}(\Omega)\\ \varphi\ne 0}}J_p(\varphi)$, as $p\to 1^+$, that is: \begin{equation*}\Lambda(\Omega,\beta)=\inf_{\substack{\varphi \in BV(\Omega)\\\varphi\not\equiv 0}}\dfrac{|Du|_F(\Omega)+\min\{\beta,1\}\displaystyle\int_{\partial \Omega}|\varphi|F(\nu)d\mathcal H^{N-1}}{\displaystyles\int_\Omega |\varphi|dx}. \end{equation*} Furthermore, for $\beta>-1$, weobtain an isoperimetric inequality for $\Lambda(\Omega,\beta)$ depending on$\beta$.  The proof uses an interior approximation result for $BV(\Omega)$ functions by$C^\infty(\Omega)$ functions in the sense of strict convergence on $\mathbbR^n$ and a trace inequality in $BV$ with respect to the anisotropic totalvariation.

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