【摘要】 设n维欧氏空间中超平面L₁与n—1维单位球面交于圆Γ₁,Γ₂为S^n-1中的满足一定条件的n—2维光滑流形,Γ₁,Γ₂不相交,所有Γ₁和Γ₂之间单位球面上的点构成S^n-1中的环面.本文利用移动平面法,得到若Γ₁由环面大侧向小侧旋转,环面上满足Dirichlet边界条件的Laplace算子的第一特征值逐渐减小.本文将Γ₂推广为满足一定条件的n-2维光滑流形.最后,本文给出了若Γ₂在一定条件约束下,当r₂固定且环面面积不变时,第一特征值取得最大值时Γ₁应该满足的条件.更多还原

【Abstract】 Assume that Γ₁ is a circle on the unit sphere S^n-1 intersected by the hyperplanes L₁ in n dimensional Euclidean space,Γ₂ is a n-2 dimensional smooth manifold satisfying certain conditions,which is disjoint with Γ₁.All the points between Γ₁ and Γ₂ on the unit sphere make up an spherical annulus.We show that the first Dirichlet eigenvalue of spherical Laplacian on a annulus decreases when Γ₁ rotates from the big side of annulus to the small side by using the moving plane method.This paper generalizes Γ₂ to the case of n-2 dimensional smooth manifold satisfying certain conditions.In the last,if Γ₂ is under certain constraints,Γ₂ is fixed and the area of annulus remains a constant as Γ₁ changes,we give the condition of Γ₁ for which the first eigenvalue gets its maximum.更多还原