On the $Γ$-limit of weighted fractional energies

TL;DR

The article establishes a Γ-convergence result for the weighted fractional p-seminorm as s→1− on bounded Lipschitz domains with uniformly converging weights f_s→f. The Γ-limit is the weighted Dirichlet p-energy with weight f^2 (and the total variation analogue for p=1), extending BBM-type limits to a nonuniform, weighted setting. For the Hilbertian (p=2) case, the authors also prove the convergence of the corresponding gradient flows, leveraging an abstract Γ-convergence framework in Hilbert spaces. The results unify and extend known unweighted Γ-convergence results to a weighted, possibly non-isotropic context and provide tools for analyzing associated parabolic problems under weight convergence.

Abstract

Given $p\in[1,\infty)$ and a bounded open set $Ω\subset\mathbb R^d$ with Lipschitz boundary, we study the $Γ$-convergence of the weighted fractional seminorm \[ [u]_{s,p,f}^p = \int_{\mathbb R^d} \int_{\mathbb R^d} \frac{|\tilde{u}(x)- \tilde{u}(y)|^p}{\|x-y\|^{d+sp}}\,f(x)\,f(y)\,\mathrm{d} x\,\mathrm{d} y \] as $s\to1^-$ for $u\in L^p(Ω)$, where $\tilde{u}=u$ on $Ω$ and $\tilde{u}=0$ on $\mathbb R^d\setminusΩ$. Assuming that $(f_s)_{s\in(0,1)}\subset L^\infty(\mathbb R^d;[0,\infty))$ and $f\in\mathrm{Lip}_b(\mathbb R^d;(0,\infty))$ are such that $f_s\to f$ in $L^\infty(\mathbb R^d)$ as $s\to1^-$, we show that $(1-s)[u]_{s,p,f_s}$ $Γ$-converges to the Dirichlet $p$-energy weighted by $f^2$. In the case $p=2$, we also prove the convergence of the corresponding gradient flows.

Theorems & Definitions (27)

• Theorem 1.1: $\Gamma$-convergence with weights
• Theorem 1.2: Stability of parabolic flows
• Theorem 2.1
• Theorem 2.2
• Theorem 3.1
• Proposition 3.2
• proof
• Proposition 3.3
• Lemma 3.4
• proof : Proof of \ref{['prop2enunciato']}