Approximation of the least Rayleigh quotient for degree $p$ homogeneous functionals
Approximation of the least Rayleigh quotient for degree $p$ homogeneous functionals
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$p$ 次齐次泛函的最小瑞利商的近似
DOI：
10.1016/j.jfa.2017.02.024
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发表时间：
2016
期刊：
影响因子：
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通讯作者：
E. Lindgren
中科院分区：
文献类型：
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作者：
Ryan Hynd;E. Lindgren
给出了两种新的逼近抽象Rayleigh商Φ（u）/φ u φ p极小的方法，其中Φ是Banach空间上的严格凸泛函，范数为φ u，且假设Φ是p∈（1，∞）次正齐次的.证明了极小元满足<$Φ（u）− λ J p（u）<$0，其中J p是1 p <$p的次微分.第一逼近方案基于方阵的逆迭代，涉及满足<$Φ（uk）− J p（uk − 1）<$0（k∈ N）的序列.第二种方法是基于双非线性演化方程Jp（vstec（t））+<$Φ（v（t））<$0（a. e. t> 0）和更一般的Φ的最大斜率的p曲线。我们证明了这两种格式都具有Rayleigh商沿沿着解非增和适当尺度的解收敛于Φ（u）/φ u_p的极小值的显著性质.这些结果即使在Hilbert空间中也是新的，它们的主要应用是Sobolev空间中不等式的最优常数和极值函数的逼近.
We present two novel methods for approximating minimizers of the abstract Rayleigh quotient Φ (u)/‖ u‖ p. Here Φ is a strictly convex functional on a Banach space with norm‖⋅‖, and Φ is assumed to be positively homogeneous of degree p∈(1,∞). Minimizers are shown to satisfy∂ Φ (u)− λ J p (u)∋ 0 for a certain λ∈ R, where J p is the subdifferential of 1 p‖⋅‖ p. The first approximation scheme is based on inverse iteration for square matrices and involves sequences that satisfy∂ Φ (u k)− J p (u k− 1)∋ 0 (k∈ N). The second method is based on the large time behavior of solutions of the doubly nonlinear evolution J p (v˙(t))+∂ Φ (v (t))∋ 0 (a. e. t> 0) and more generally p-curves of maximal slope for Φ. We show that both schemes have the remarkable property that the Rayleigh quotient is nonincreasing along solutions and that properly scaled solutions converge to a minimizer of Φ (u)/‖ u‖ p. These results are new even for Hilbert spaces and their primary application is in the approximation of optimal constants and extremal functions for inequalities in Sobolev spaces.