$\Gamma$-Convergence of Weak-Type Nonlocal Functionals on Bounded Domains
Let $N\ge1$, $p\in[1,\infty)$, $\gamma\in(0,\infty)$, and $\Omega\subset\mathbb R^N$ be a bounded open interval when $N=1$ or a bounded Lipschitz domain when $N\ge2$. For any $\lambda\in(0,\infty)$ and any measurable function $u$, consider the weak-type nonlocal functional \begin{align*} G_{\lambda,p,\gamma}(u;\Omega) :=\lambda\iint_{\Omega\times\Omega} \mathbf 1_{\left\{(x,y)\in\Omega\times\Omega:\ x\neq y,\ \frac{|u(x)-u(y)|^p}{|x-y|^{p+\gamma}}\geq\lambda\right\}} |x-y|^{\gamma-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*} \Psi_{p,\gamma}^{\mathrm{cell}}(u;\Omega):= \begin{cases} C_{N,p,\gamma}^{\mathrm{cell}}\displaystyle\int_\Omega|\nabla u|^p\,dx, &p\in(1,\infty)\ \hbox{and}\ u\in W^{1,p}(\Omega),\\[2mm] C_{N,1,\gamma}^{\mathrm{cell}}|Du|(\Omega), &p=1\ \hbox{and}\ u\in BV(\Omega),\\[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].