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

Abstract

Let $N\ge 1$, $p\in [1,\infty )$, $\gamma \in (0,\infty )$, 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 $\lambda \in (0,\infty )$ 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 $\lambda \to \infty$, the family $G_{\lambda ,p,\gamma }$ converges, in the sense of $\Gamma$-convergence in $L^p(\Omega )$, to the functional \begin{align*} Ψ_{p,γ}^{\mathrm{cell}}(u;Ω):= \begin{cases} C_{N,p,γ}^{\mathrm{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}^{\mathrm{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 $\Omega$ 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].

42 pages