preprint

Inserted: 18 oct 2026

Year: 2023

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 \[ J_p(\varphi)=\frac{\int_ΩF^p(\nabla \varphi)dx+β\int_{\partial Ω}
\varphi
^pF(ν)d\mathcal{H}^{N-1}}{\int_Ω
\varphi
^pdx} \] where $\varphi\in W^{1,p}(Ω)\setminus\{0\}$ and $F$ is a sufficientely smooth norm on $\mathbb R^n$. We study the limit of the first eigenvalue $λ_1(Ω,p,β)=\inf_{\substack{\varphi\in W^{1,p}(Ω)\\ \varphi \ne 0}}J_p(\varphi)$, 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 $β>-1$, 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.