截断函数及在偏微分方程中的应用  

Cut-ofF Function and Its Applications in Partial Differential Equations

作　　者：吴佩英 WU Peiying(Mudanqu No.1 Primary School,Heze 274000,China)

机构地区：[1]菏泽市牡丹区第一小学,菏泽274000

出　　处：《应用数学学报》2021年第6期801-806,共6页Acta Mathematicae Applicatae Sinica

摘　　要：在偏微分方程中,引进截断函数是使问题局部化的一种重要手段,这样既可以完整地保留被切断函数的局部性质,又可以有效地避免小邻域以外各种因素的影响.因此,本文通过引入磨光算子,利用磨光算子的定义和性质,并结合Holder不等式,证明了截断函数的一个较为重要的性质,即截断函数的任意阶导数与本身p(1<p<∞)次幂的比值为一个常数,此常数仅与导数的阶数和p有关.In partial differential equations,the introduction of cut-off function is an im-portant mean to localize the problem,which can not only preserve the local property ofthe truncated function,but also effectively avoid the influence of various factors outside thesmall neighborhood.In this paper,we first introduce the mollification,then an importantproperty of cut-off functions are proved by using the definition and properties of the stan-dard mollifier,together with the Holder inequality.More precisely,we show that the ratioof any order derivatives of a cut-off function to the p-th power of itself is a constant wherethe constant depends only on the order of derivatives and p.

关 键 词：偏微分方程 磨光算子 截断函数 HOLDER不等式 

分 类 号：O212.7[理学—概率论与数理统计]