Full-Range $\Gamma$-Convergence of Weak-Type Nonlocal Functionals

Abstract

Let $N\ge 1$, $p\in [1,\infty )$, and $\Omega \subset {\mathbb{R}}^N$ be a bounded open interval when $N=1$ or a bounded Lipschitz domain when $N\ge 2$. For any given $\gamma \in \mathbb{R}\setminus \{0\}$ and $\lambda \in (0,\infty )$, 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 $\gamma \in (0,\infty )$ as $\lambda \to \infty$ or for $\gamma \in (-\infty ,-1]$ as $\lambda \to 0^{+}$, the family $F_{\lambda ,p,\gamma }(\cdot ;\Omega )$ converges, in the sense of $\Gamma$-convergence in $L^p(\Omega )$, 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 $C_{N,p,\gamma }^{\mathrm{cell}}$ are independent of $\Omega$. As applications, we further show that, for $\gamma \in (0,\infty )$ as $\lambda \to \infty$ or for $\gamma \in (-\infty ,-1]$ as $\lambda \to 0^{+}$, the weak-type nonlocal functional $F_{\lambda ,p,\gamma }(\cdot ;{\mathbb{R}}^N)$ converges, in the sense of $\Gamma$-convergence in $L_{\mathrm{loc}}^1({\mathbb{R}}^N)$, to a similar functional; this gives an complete answer to the problem posed by Brezis et al. in [Section~7C, Anal. PDE 2024].