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Article overview
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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 | Abstract: | Let $Omega$ 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 | Services: | Forum | Review | PDF | Favorites |
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