Full-Range $Γ$-Convergence of Weak-Type Nonlocal Functionals

Abstract

Let $N\ge1$, $p\in[1,\infty)$, and $Ω\subset\mathbb R^N$ be a bounded open interval when $N=1$ or a bounded Lipschitz domain when $N\ge2$. For any given $γ\in\mathbb R\setminus\{0\}$ and $λ\in(0,\infty)$, consider the weak-type nonlocal functional \begin{align*} F_{λ,p,γ}(u;Ω):=λ^p \underset{|u(x)-u(y)|>λ|x-y|^{1+\fracγ{p}}} {\int_Ω\int_Ω}|x-y|^{γ-N}\,dy\,dx. \end{align*} In this article, we prove that, for $γ\in(0,\infty)$ as $λ\to\infty$ or for $γ\in(-\infty,-1]$ as $λ\to0^+$, the family $F_{λ,p,γ}(\cdot;Ω)$ converges, in the sense of $Γ$-convergence in $L^p(Ω)$, to the functional \begin{align*} Ψ_{p,γ}^Ω(u) :=\begin{cases} C_{N,p,γ}^{\rm cell}\displaystyle\int_Ω|\nabla u|^p\,dx, &p\in(1,\infty)\ \hbox{and}\ u\in W^{1,p}(Ω),\\[2mm] C_{N,1,γ}^{\rm cell}|Du|(Ω), &p=1\ \hbox{and}\ u\in BV(Ω),\\[1mm] \infty,&\hbox{otherwise}. \end{cases} \end{align*} where the positive constants $C_{N,p,γ}^{\mathrm{cell}}$ are independent of $Ω$. As applications, we further show that, for $γ\in(0,\infty)$ as $λ\to\infty$ or for $γ\in(-\infty,-1]$ as $λ\to0^+$, the weak-type nonlocal functional $F_{λ,p,γ}(\cdot;\mathbb R^N)$ converges, in the sense of $Γ$-convergence in $L^1_{\rm loc}(\mathbb R^N)$, to a similar functional; this gives an complete answer to the problem posed by Brezis et al. in [Section~7C, Anal. PDE 2024].