对数各向异性Sobolev不等式

冯廷福,董　艳

(西北工业大学 应用数学系,陕西 西安 710129)



对数各向异性Sobolev不等式

冯廷福,董艳

(西北工业大学 应用数学系,陕西 西安 710129)

摘要:利用Hölder不等式,分别结合各向异性Sobolev不等式和带权各向异性Sobolev不等式,得到了对数各向异性Sobolev不等式和对数带权各向异性Sobolev不等式, 从而将对数Sobolev不等式推广到对数各向异性情形.

关键词:Hölder不等式; 对数各向异性Sobolev不等式; 对数带权各向异性Sobolev不等式

1引言和主要结果

各向异性椭圆方程来自各向异性介质的物理性质研究[1-4].近年来,已有学者研究了各向异性椭圆方程解的可积性和有界性等, 其中各向异性Sobolev不等式起着重要作用[3-6]. 注意到Merker[7]利用Hölder不等式证明了如下形式的对数各向同性Sobolev不等式

(1)

成立.并可得

注若1≤pi=pmax=p<+∞(i=1,2,…,n), 则定理1中的对数各向异性Sobolev不等式就成为对数各向同性Sobolev不等式.

进一步有

注当权函数νi=1(i=1,2,…,n)时, 由定理2即得定理1中当1

2定理1的证明

各向异性Sobolev空间形如

W1,(pi)(Ω)={u∈W1,1(Ω):Diu∈Lpi(Ω),i=1,…,n},

其范数分别为

(2)

注意到

从而由式(2)可推出

下面利用Hölder不等式证明定理1.

(3)

(4)

则由式(4)知函数φ:1/r→log(‖u‖Lr)在[0,+∞)是凸函数, 即

(5)

且

(6)

由于φ在[0,+∞)的凸性等价于

(7)

(8)

因为

(9)

则式(8)中的第一项乘以式(9)可得

(10)

由式(10)和引理1可得对数各向异性Sobolev不等式.定理1得证.

3定理2的证明

对于每个10,令

那么带权的各向异性空间形如

W1,(pi)(Ω,νi)={u∈W1,1(Ω):νi|Diu|pi∈L1(Ω),i=1,…,n},

其范数分别为

(11)

其中u∈Lpmax(Ω)∩Lpm(Ω).现在使用定理1中的证明过程,就可由(11)得不等式

(12)

再由式(12)和引理2即得对数带权各向异性Sobolev不等式.定理2得证.

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编辑、校对：师琅

文章编号:1006-8341(2016)02-0166-05

DOI：10.13338/j.issn.1006-8341.2016.02.006

收稿日期：2015-08-23

基金项目:国家自然科学基金资助项目(11271299)

通讯作者:冯廷福(1986—),男，云南省大理白族自治州人,西北工业大学博士研究生,研究方向为偏微分方程及其应用.E-mail:ftfml@mail.nwpu.edu.cn

中图分类号：O 178

文献标识码：A

Logarithmic anisotropic Sobolev inequalities

FENGTingfu,DONGYan

(Department of Applied Mathematics, Northwestern Polytechnical University, Xi′an 710129, China)

Abstract:The logarithmic anisotropic Sobolev inequality is proven by using the Hölder inequality combine with the anisotropic Sobolev inequality. Furthermore, the logarithmic weighted anisotropic Sobolev inequality is obtained by the same way. It generalizes logarithmic Sobolev inequalities to the logarithmic anisotropic case.

Key words:Hölder inequality;logarithmic anisotropic Sobolev inequality;logarithmic weighted anisotropic Sobolev inequality

引文格式：冯廷福,董艳.对数各向异性Sobolev不等式[J].纺织高校基础科学学报,2016,29(2)：166-170.

FENG Tingfu,DONG Yan.Logarithmic anisotropic Sobolev inequalities[J].Basic Sciences Journal of Textile Universities,2016,29(2):166-170.