Neumann边界条件下Lp-Poincaré不等式最优常数的估计

靳荷艳

数学学报 ›› 2014, Vol. 57 ›› Issue (4) : 665-674.

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数学学报 ›› 2014, Vol. 57 ›› Issue (4) : 665-674. DOI: 10.12386/A2014sxxb0061
论文

Neumann边界条件下Lp-Poincaré不等式最优常数的估计

    靳荷艳
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Estimation of the Optimal Constants in the Lp-Poincaré Inequalities under Neumann Boundary Condition

    He Yan JIN
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摘要

本文考虑在Neumann 边界条件下，当π 为有限测度时，对不等式 π（|f-π（|f|p-2f）|p）≤āpπ（a|f'|p），f∈D（Dp）中的最优常数āp的估计. 通过采用分割的方法可转化 为Dirichlet 边界条件的情况，进而得到了上下界的估计. 并考虑当π为无穷测度时，在 Neumann 边界条件下不等式 π（|f|p）≤?pπ（a|f'|p），f∈D（Dp） 中常数?p的上下界，给出了变分公式估计及显式估计.

Abstract

We consider in this paper the estimation of the optimal constant āp in π(|f-π(|f|p-2f)|p)≤āpπ(a|f'|p)，f∈D(Dp) under Neumann boundary condition when π is a finite measure. We can get the estimations of upper and lower bounds by using cutting method to transform this problem to Dirichlet boundary condition. When π is infinite measure, we consider the upper and lower bound of the constant in π(|f|p)≤?pπ(a|f'|p),f∈D(Dp) under Neumann boundary condition. We give the variational formula and the explicit estimation.

关键词

Lp-Poincaré / 不等式 / Neumann 边界条件 / 变分公式 / 显式界

Key words

Lp-Poincaré / inequality / Neumann boundary condition / variational formula / explicit boundary

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靳荷艳. Neumann边界条件下Lp-Poincaré不等式最优常数的估计. 数学学报, 2014, 57(4): 665-674 https://doi.org/10.12386/A2014sxxb0061
He Yan JIN. Estimation of the Optimal Constants in the Lp-Poincaré Inequalities under Neumann Boundary Condition. Acta Mathematica Sinica, Chinese Series, 2014, 57(4): 665-674 https://doi.org/10.12386/A2014sxxb0061

参考文献

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