Extremal values of eigenvalues of Sturm-Liouville operators with potentials in L 1 balls
Extremal values of eigenvalues of Sturm-Liouville operators with potentials in L 1 balls
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DOI：
10.1016/j.jde.2009.04.008
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发表时间：
2009-07
影响因子：
2.4
通讯作者：
Qiaoling Wei;Gang Meng;Meirong Zhang
中科院分区：
文献类型：
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作者：
Qiaoling Wei;Gang Meng;Meirong Zhang
本文是张[M.]张，弱拓扑的连续性：ODE的高阶线性系统，Sci。中国爵士。A 51 (2008) 1036-1058；张明。一类具有势的Hill算子的最小特征值的极值[j].微分方程[j](2009): 488 - 492。给定一个势q∈Lp([0,1]，R), p∈[1,∞]。我们用λm(q)表示势为q(t)的Sturm-Liouville算子的第m个Dirichlet特征值，其中m∈N。λm(q)的最小值Lm，p(r)和最大值Mm，p(r)在半径为r的Lpball中具有势q。本文利用弱拓扑中λm(q)在q中的连续性，利用变分方法给出了当p∈(1,∞)时Lm，p(r)和Mm，p(r)的刻画。利用p↓1的极限方法，我们发现最重要的极值Lm，1(r)和Mm，1(r)可以用r的一些初等函数显式求值。相应的Neumann特征值和一些周期特征值的极值问题将被简化为Lm，p(r)和Mm，p(r)。
This paper is a continuation of Zhang [M. Zhang, Continuity in weak topology: Higher order linear systems of ODE, Sci. China Ser. A 51 (2008) 1036–1058; M. Zhang, Extremal values of smallest eigenvalues of Hill's operators with potentials in L1balls, J. Differential Equations 246 (2009) 4188–4220]. Given a potential q∈Lp([0,1],R), p∈[1,∞]. We use λm(q) to denote the mth Dirichlet eigenvalue of the Sturm–Liouville operator with potential q(t), where m∈N. The minimal value Lm,p(r) and the maximal value Mm,p(r) of λm(q) with potentials q in the Lpball of radius r are well defined. In this paper, we will exploit the continuity of λm(q) in q with weak topologies and the variational method to give characterizations of Lm,p(r) and Mm,p(r) when p∈(1,∞). By using the limiting approach as p↓1, we find that the most important extremal values Lm,1(r) and Mm,1(r) can be evaluated explicitly using some elementary functions of r. The corresponding extremal problems for Neumann eigenvalues and some periodic eigenvalues will be reduced to Lm,p(r) and Mm,p(r).