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

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 $p\to 1^{+}$, 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 $\phi \in W^{1,p}(\Omega )\setminus \{0\}$ and $F$ is a sufficientely smooth norm on ${\mathbb{R}}^n$. We study the limit of the first eigenvalue ${\lambda }_1(\Omega ,p,\beta )={\inf }_{\begin{array}{c}\phi \in W^{1,p}(\Omega )\\ \phi \ne 0\end{array}}{J}_p(\phi )$, 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 $\beta >-1$, 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.

References in corpus (1)

• Sharp estimates for the first Robin eigenvalue of nonlinear elliptic operators

Cited by in corpus (1)

• A stability result for the first Robin-Neumann eigenvalue: A double perturbation approach