一维p-Laplace方程前两个特征值的比值问题

Problems of Eigenvalue Ratios for One-dimensional p-Laplace Equations

本文研究了一维p-Laplace问题■的前两个特征值的比值,其中ρ(x)是满足不等式1≤ρ(x)≤H的分段连续函数,找到了使比值λ_2/λ_1取得最小值的密度函数ρ_0(x).此外我们研究了比值λ_2/λ_1关于分段函数ρ(x) ={H,x∈[0.a] 1,x∈(a,1]的间断点a的变化情况.

The ratio of the two eigenvalues of one-dimensional p-Laplace problem {-（｜u′（x）｜^p-2u′（x）′=（p-1）λρ（x）｜u（x）｜^p-2u（x） u（0）=u（1）=0 is investigated,where ρ（x） is piecewise continuous and satisfying the inequality 1≤ρ（x）≤H,the extremizing density which provide the min value for the ratio（λ2）/（λ1） is found.And giving the 1-step density ρ（x） ={H,x∈[0.a] 1,x∈（a,1] we are concerned with the changing of the ratio（λ2）/（λ） with respect to the jump a.