弱型非局部泛函的全范围 $\Gamma$-收敛
Full-Range $Γ$-Convergence of Weak-Type Nonlocal Functionals
AI总结：
本文证明弱型非局部泛函在特定参数范围下按 $\Gamma$-收敛趋于局部梯度泛函，并应用于全空间情形，完整解答了 Brezis 等人提出的问题。
AI中文摘要：
设 $N\ge1$，$p\in[1,\infty)$，且当 $N=1$ 时 $\Omega\subset\mathbb R^N$ 为有界开区间，当 $N\ge2$ 时为有界 Lipschitz 域。对任意给定的 $\gamma\in\mathbb R\setminus\{0\}$ 和 $\lambda\in(0,\infty)$，考虑弱型非局部泛函 \begin{align*} F_{\lambda,p,\gamma}(u;\Omega):=\lambda^p \underset{|u(x)-u(y)|>\lambda|x-y|^{1+\frac{\gamma}{p}}} {\int_\Omega\int_\Omega}|x-y|^{\gamma-N}\\,dy\\,dx. \end{align*} 本文证明，当 $\gamma\in(0,\infty)$ 且 $\lambda\to\infty$ 时，或当 $\gamma\in(-\infty,-1]$ 且 $\lambda\to0^+$ 时，族 $F_{\lambda,p,\gamma}(\cdot;\Omega)$ 在 $L^p(\Omega)$ 中按 $\Gamma$-收敛的意义收敛到泛函 \begin{align*} \Psi_{p,\gamma}^{\Omega}(u):=\begin{cases} C_{N,p,\gamma}^{\rm cell}\displaystyle\int_\Omega|\nabla u|^p\\,dx, &p\in(1,\infty)\\ \hbox{且}\\ u\in W^{1,p}(\Omega),\\\\[2mm] C_{N,1,\gamma}^{\rm cell}|Du|(\Omega), &p=1\\ \hbox{且}\\ u\in BV(\Omega),\\\\[1mm] \infty,&\hbox{其他情况}. \end{cases} \end{align*} 其中正常数 $C_{N,p,\gamma}^{\mathrm{cell}}$ 与 $\Omega$ 无关。作为应用，我们进一步证明，当 $\gamma\in(0,\infty)$ 且 $\lambda\to\infty$ 时，或当 $\gamma\in(-\infty,-1]$ 且 $\lambda\to0^+$ 时，弱型非局部泛函 $F_{\lambda,p,\gamma}(\cdot;\mathbb R^N)$ 在 $L^1_{\rm loc}(\mathbb R^N)$ 中按 $\Gamma$-收敛的意义收敛到类似泛函；这完整回答了 Brezis 等人在 [Section~7C, Anal. PDE 2024] 中提出的问题。
英文摘要：
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].
