$\mathrm{\Gamma }$-Convergence of Weak-Type Nonlocal Functionals on Bounded Domains

Let $N\ge 1$, $p\in [1,\mathrm{\infty })$, $\gamma \in (0,\mathrm{\infty })$, and $\mathrm{\Omega }\subset {\mathbb{R}}^N$ be a bounded open interval when $N=1$ or a bounded Lipschitz domain when $N\ge 2$. For any $\lambda \in (0,\mathrm{\infty })$ and any measurable function $u$, consider the weak-type nonlocal functional \begin{align*} G_{λ,p,γ}(u;Ω) :=λ\iint_{Ω\timesΩ} \mathbf 1_{\left{(x,y)\inΩ\timesΩ:\ x\neq y,\ \frac{|u(x)-u(y)|^p}{|x-y|^{p+γ}}\geqλ\right}} |x-y|^{γ-N},dx,dy. \end{align*} In this article, we prove that, as $\lambda \to \mathrm{\infty }$, the family $G_{\lambda ,p,\gamma }$ converges, in the sense of $\mathrm{\Gamma }$-convergence in $L^p(\mathrm{\Omega })$, to the functional \begin{align*} Ψ_{p,γ}^{$\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}$}(u;Ω):= \begin{cases} C_{N,p,γ}^{$\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}$}\displaystyle\int_Ω|\nabla u|^p,dx, &p\in(1,\infty)\ \hbox{and}\ u\in W^{1,p}(Ω),\[2mm] C_{N,1,γ}^{$\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}$}|Du|(Ω), &p=1\ \hbox{and}\ u\in BV(Ω),\[1mm] \infty,&\hbox{otherwise}, \end{cases} \end{align*} where the positive constants $C_{N,p,\gamma }^{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}$ are independent of $\mathrm{\Omega }$ and characterized by a cell formula. This gives an affirmative answer to the problem posed by Brezis [Open Problem~9.3, Rend. Lincei Mat. Appl. 2023].