AI 中文总结
该研究针对Nguyen和Squassina提出的开放问题，通过构造显式反例证明非局部阈值能量在施瓦茨重排下不会减小，还计算了一维能量间隙闭式。
AI 中文摘要
对于N≥1、p≥1且δ>0，考虑非局部阈值泛函I_{δ,p}(u)=∬_{{|u(x)-u(y)|>δ}} δ^p/|x-y|^{N+p} dxdy。Nguyen和Squassina提出，I_{δ,p}在施瓦茨重排下是否会减小？我们在所有维度中给出否定答案。对于每个N≥1和δ>0，存在一个对所有p≥1均适用的显式反例：该反例非负、有界、紧支且仅取五个值，其四个非平凡上水平集为嵌套球，球心在两点间交替。分离活跃阈值相互作用后，能量差可简化为单位球与同心环、偏心壳的相互作用比较，因单位球产生的势随半径减小，故符号严格为负。我们还计算了一维下的能量间隙闭式，解决了Nguyen和Squassina的开放问题2.2。
英文摘要
For $N\geq 1$, $p\geq 1$, and $δ>0$, consider the nonlocal threshold functional \[ I_{δ,p}(u)=\iint_{\{|u(x)-u(y)|>δ\}} \frac{δ^p}{|x-y|^{N+p}}\,\dd x\,\dd y. \] Nguyen and Squassina \cite{NguyenSquassina} asked whether $I_{δ,p}$ decreases under Schwarz rearrangement. We give a negative answer in every dimension. For each $N\geq1$ and $δ>0$, one and the same explicit counterexample works for every $p\geq1$: it is nonnegative, bounded, compactly supported, and takes only five values. Its four nontrivial superlevel sets are nested balls whose centers alternate between two points. Once the active threshold interactions are isolated, the energy difference reduces to comparing the interaction of a unit ball with a concentric annulus and with an eccentric shell. The sign is strict because the potential generated by the unit ball decreases with the radius. We also compute the energy gap in closed form in dimension one. This settles Open Problem~2.2 of Nguyen and Squassina.
