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07 February 2025
 
  » arxiv » 2301.01546

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On the first Robin eigenvalue of the Finsler $p$-Laplace operator as $p o 1$
Rosa Barbato ; Francesco Della Pietra ; Gianpaolo Piscitelli ;
Date 4 Jan 2023
AbstractLet $Omega$ be a bounded, connected, sufficiently smooth open set, $p>1$ and $etainmathbb R$. In this paper, we study the $Gamma$-convergence, as $p ightarrow 1^+$, of the functional [ J_p(varphi)=frac{int_Omega F^p( abla varphi)dx+etaint_{partial Omega} |varphi|^pF( u)dmathcal{H}^{N-1}}{int_Omega |varphi|^pdx} ]
where $varphiin W^{1,p}(Omega)setminus{0}$ and $F$ is a sufficientely smooth norm on $mathbb R^n$. We study the limit of the first eigenvalue $lambda_1(Omega,p,eta)=inf_{substack{varphiin W^{1,p}(Omega)\ varphi e 0}}J_p(varphi)$, as $p o 1^+$, that is: egin{equation*} Lambda(Omega,eta)=inf_{substack{varphi in BV(Omega)\ varphi otequiv 0}}dfrac{|Du|_F(Omega)+min{eta,1}displaystyle int_{partial Omega}|varphi|F( u)dmathcal H^{N-1}}{displaystyle sint_Omega |varphi|dx}. end{equation*} Furthermore, for $eta>-1$, we obtain an isoperimetric inequality for $Lambda(Omega,eta)$ depending on $eta$.
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.
Source arXiv, 2301.01546
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