$p$次齐次泛函的最小Rayleigh商的逼近
Approximation of the least Rayleigh quotient for degree $p$ homogeneous functionals
摘要：
我们提出了两种新的方法来逼近抽象瑞利商$Φ(u)/ \|u\|^p$的极小。这里$Φ$是Banach空间上的一个严格凸泛函，其范数为$\|\cdot\|$，并且$Φ$是$p\in (1,\infty)$次正齐次的.极小化满足$\partial Φ(u)- λ\mathcal{J}_p(u)\ni 0$为一定的$λ\in \mathbb{R}$，其中$\mathcal{J}_p$是次微分的$\frac{1}{p}\|\cdot\|^p$。第一种近似方法是基于方阵的逆迭代，涉及满足$$ \partial Φ(u_k)- \mathcal{J}_p(u_{k-1})\ni 0 \quad (k\in \mathbb{N}). $$的序列。第二种方法是基于双非线性演化$$ \mathcal{J}_p(\dot v(t))+\partialΦ(v(t))\ni 0 \quad(a.e.\;t>0) $$和更一般的$p$ -$Φ$的最大斜率曲线的解的大时间行为。我们表明，这两个计划有显着的性质，瑞利商是不增加沿解决方案和适当缩放的解决方案收敛到一个极小的$Φ(u)/ \|u\|^p$。这些结果是新的，甚至希尔伯特空间和他们的主要应用是在逼近的最佳常数和极值函数不等式在Sobolev空间。
英文摘要：
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 $\|\cdot\|$, and $Φ$ is assumed to be positively homogeneous of degree $p\in (1,\infty)$. Minimizers are shown to satisfy $\partial Φ(u)- λ\mathcal{J}_p(u)\ni 0$ for a certain $λ\in \mathbb{R}$, where $\mathcal{J}_p$ is the subdifferential of $\frac{1}{p}\|\cdot\|^p$. The first approximation scheme is based on inverse iteration for square matrices and involves sequences that satisfy $$ \partial Φ(u_k)- \mathcal{J}_p(u_{k-1})\ni 0 \quad (k\in \mathbb{N}). $$ The second method is based on the large time behavior of solutions of the doubly nonlinear evolution $$ \mathcal{J}_p(\dot v(t))+\partialΦ(v(t))\ni 0 \quad(a.e.\;t>0) $$ 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.
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