preprint

Inserted: 2 aug 2023

Year: 2023

Abstract:

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 \[ J_p(\varphi)=\frac{\int_\Omega F^p(\nabla \varphi)dx+\beta\int_{\partial \Omega}
\varphi
^pF(\nu)d\mathcal{H}^{N-1}}{\int_\Omega
\varphi
^pdx} \] where $\varphi\in W^{1,p}(\Omega)\setminus\{0\}$ and $F$ is a sufficientely smooth 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}}{\displaystyle s\int_\Omega
\varphi
dx}. \end{equation} Furthermore, for $\beta>-1$, we obtain 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 $\mathbb R^n$ and a trace inequality in $BV$ with respect to the anisotropic total variation.