Sharp Uncertainty Principle inequality for solenoidal fields
Sharp Uncertainty Principle inequality for solenoidal fields
复制标题
螺线管磁场的尖锐不确定性原理不等式
DOI：
10.1016/j.matpur.2023.01.008
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发表时间：
2023
影响因子：
2.3
通讯作者：
Hamamoto Naoki
中科院分区：
文献类型：
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作者：
Osada Hirofumi;Osada Shota;長田翔太;長田翔太;長田翔太;Koike Kai;Koike Kai;Kai Koike;Kai Koike;小池開;Kai Koike;Kai Koike;小池開;Kai Koike;Hamamoto Naoki
本文解决了∫RN|∇u|2d x∫RN|u|2 dx≥RN|u|2|x∫RN|u|2 dx∫CN(≥RN|u|2 dx)2关于RN上无散度向量场u=u(X)的尖锐不确定原理的公开问题[1，第3.9节]。N∫3的常量的最佳值是CN=1 4(N 2≥4(N−3)+2)2，超过了无约束场的初始值N2/4。此外，我们还证明了C-N的可达性，并给出了极值螺线管场的分布：对于N≥4，极值与轴对称且在对称轴上唯一的极向场成正比；当N=3时，除了极值极向场外，还存在极值环向场；当N=2时，极值场都是环向场。
This paper solves the L 2 version of Maz'ya's open problem [1, Section 3.9] on the sharp uncertainty principle inequality∫ R N|∇ u| 2 d x∫ R N| u| 2| x| 2 d x≥ C N (∫ R N| u| 2 d x) 2 for solenoidal (namely divergence-free) vector fields u= u (x) on R N. The best value of the constant for N≥ 3 turns out to be C N= 1 4 (N 2− 4 (N− 3)+ 2) 2 which exceeds the original value N 2/4 for unconstrained fields. Moreover, we show the attainability of C N and specify the profiles of the extremal solenoidal fields: for N≥ 4, the extremals are proportional to a poloidal field that is axisymmetric and unique up to the axis of symmetry; for N= 3, there exist extremal toroidal fields, in addition to extremal poloidal fields; for N= 2, the extremal fields are all toroidal.