Y-划分是圆盘和谐振子的最优三分划
The Y-partition is the optimal three-partition for the disc and the harmonic oscillator
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中文总结 AI 辅助
本文证明Y-划分是圆盘Dirichlet拉普拉斯和平面谐振子的最优三分划，最小能量分别为$j_{3/2,1}^{2}$和$5$，证实了Helffer-Hoffmann-Ostenhof猜想，方法为正径向移植结合三月瓣定理。
中文摘要 AI 辅助
我们证明了将单位圆盘上的Dirichlet拉普拉斯算子以及平面谐振子$-\Delta+|x|^2$划分为三个相等扇区的Y-划分是最小谱三分划，其最小能量分别为$j_{3/2,1}^{2}$和$5$；对于圆盘，这证实了Helffer和Hoffmann-Ostenhof的一个猜想。最小化正则强划分在旋转意义下是唯一的，并且每个开最小化划分的单元都具有扇区的Dirichlet形式域。证明方法是向球面进行正径向移植，该移植保持分离性并匹配分离模型态的角能量测度；然后Helffer、Hoffmann-Ostenhof和Terracini的三月瓣定理给出了下界。该移植通过具有严格正径向权的非负缺陷降低了平移后的二次型；在等式情形下，球面等分使缺陷消失，从而分离变量，圆上的Poincaré不等式识别出扇区。
英文摘要
We prove that the Y-partition into three equal sectors is the minimal spectral three-partition both for the Dirichlet Laplacian on the unit disc and for the planar harmonic oscillator $-Δ+|x|^2$, with minimal energies $j_{3/2,1}^{2}$ and $5$; for the disc, this confirms a conjecture of Helffer and Hoffmann-Ostenhof. The minimizing regular strong partition is unique up to rotation, and every open minimizing partition has cells with the Dirichlet form domains of the sectors. The proof is a positive radial transplantation to the sphere that preserves segregation and matches the angular-energy measures of the separated model states; the three-lune theorem of Helffer, Hoffmann-Ostenhof, and Terracini then gives the lower bound. The transplantation lowers the shifted quadratic form by a nonnegative defect with strictly positive radial weight; in the equality case, spherical equipartition makes the defects vanish, which separates variables, and a Poincaré inequality on the circle identifies the sectors.
发表机构
- Lund University（隆德大学）
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