LU Kang，HUANG Zhen-you.Some Properties of the First Eigenvalue of Laplace Operator on a Spherical Band[J].Journal of Chongqing Technology and Business University(Natural Science Edition）,2021,38(2):35-39

Some Properties of the First Eigenvalue of Laplace Operator on a Spherical Band

DOI：

 作者 单位 吕康， 黄振友 南京理工大学 理学院，南京 210094

Laplace算子特征值的研究在物理上有着重要的应用，它与粒子在力场中运动时所具有的能级有密切关系，根据最大-最小原理，可以对特征值进行理论上的表示；针对三维欧式空间单位球的球带上具有Robin型边条件的Laplace算子的特征值问题，先利用Courant节点域定理和最大-最小原理，求出了第一特征值的理论表示; 然后利用此表示，证明了球带在关于赤道对称时第一特征值最大(球带面积固定); 且若球带的面积小于等于2π，有当球带向赤道靠近时，第一特征值会严格增加的结果。

The researches of the eigenvalue problems of Laplace operator have many applications to physics，they relate to the levels of power belonging to the particles when particles across certain fields.By Maximum Minimum Principle，we can formulate the eigenvalues theoretically.Based on the eigenvalue problem of Laplace operator on a spherical band in three dimensional space with Robin boundary conditions，we formulated the first eigenvalue by virtue of Courant nodal domain theorem and Maximum Minimum Principle，then by this formulation，we proved that the first eigenvalue would be maximum when the band is symmetric about the equator (the area of the band is constant)，also，if the area of the band is smaller than or equals to 2π，then the first eigenvalue would monotonously increase when the band moves to the equator.