发表机构
University of Rome Tor Vergata; University of Trento; University of Pavia（罗马托维尔加塔大学; 特伦托大学; 帕维亚大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过球面重排证明Maz'ya-Shaposhnikova极限的Γ-下极限不等式，并应用于截断能量，证明其Γ-收敛到(1-λ)Nω_N‖u‖_{L^2}^2，实现长程贡献识别与连续插值。
AI 中文摘要
我们基于球面重排，给出了当s→0+时分数阶Sobolev半范数的Maz'ya-Shaposhnikova极限的Γ-下极限不等式的一个证明。关键点在于，弱收敛序列的重排是单调径向轮廓，由Helly定理它们强收敛；由于能量在重排下（在极限中）不增加，这便将下极限不等式归结为径向递减函数的逐点收敛陈述。作为应用，我们处理截断能量，其中相互作用被限制在半径为r_s→∞的球内。假设r_s^{-2s}→λ∈[0,1]，我们证明缩放后的能量关于弱L^2收敛Γ-收敛到(1-λ)Nω_N‖u‖_{L^2}^2，其中ω_N是R^N中单位球的测度。这识别了长程相互作用的贡献，并在经典Maz'ya-Shaposhnikova极限（λ=0）与能量消失区域（λ=1）之间给出了连续插值。
英文摘要
We give a proof of the $Γ$-liminf inequality for the Maz'ya--Shaposhni\-kova limit, as $s\to0^+$, of the fractional Sobolev seminorm, based on spherical rearrangements. The key point is that the rearrangements of a weakly convergent sequence are monotone radial profiles, which converge strongly by Helly's theorem; since the energy does not increase under rearrangement (in the limit), this reduces the liminf inequality to a pointwise convergence statement for radially decreasing functions. As an application, we treat truncated energies, in which the interaction is restricted to a ball of radius $r_s\to\infty$. Assuming $r_s^{-2s}\toλ\in[0,1]$, we prove that the scaled energies $Γ$-converge, with respect to weak $L^2$-convergence, to $$ (1-λ)Nω_N\|u\|_{L^2}^2, $$ where $ω_N$ is the measure of the unit ball of $\mathbb R^N$. This identifies the contribution of the long-range interactions and gives a continuous interpolation between the classical Maz'ya--Shaposhnikova limit ($λ=0$) and the vanishing-energy regime ($λ=1$).
