對數(shù)各向異性Sobolev不等式

馮廷福,董　艷

(西北工業(yè)大學(xué) 應(yīng)用數(shù)學(xué)系,陜西 西安 710129)

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對數(shù)各向異性Sobolev不等式

馮廷福,董艷

(西北工業(yè)大學(xué) 應(yīng)用數(shù)學(xué)系,陜西 西安 710129)

摘要:利用H?lder不等式,分別結(jié)合各向異性Sobolev不等式和帶權(quán)各向異性Sobolev不等式,得到了對數(shù)各向異性Sobolev不等式和對數(shù)帶權(quán)各向異性Sobolev不等式, 從而將對數(shù)Sobolev不等式推廣到對數(shù)各向異性情形.

關(guān)鍵詞:H?lder不等式; 對數(shù)各向異性Sobolev不等式; 對數(shù)帶權(quán)各向異性Sobolev不等式

1引言和主要結(jié)果

各向異性橢圓方程來自各向異性介質(zhì)的物理性質(zhì)研究[1-4].近年來,已有學(xué)者研究了各向異性橢圓方程解的可積性和有界性等, 其中各向異性Sobolev不等式起著重要作用[3-6]. 注意到Merker[7]利用H?lder不等式證明了如下形式的對數(shù)各向同性Sobolev不等式

(1)

成立.并可得

注若1≤pi=pmax=p<+∞(i=1,2,…,n), 則定理1中的對數(shù)各向異性Sobolev不等式就成為對數(shù)各向同性Sobolev不等式.

進一步有

注當(dāng)權(quán)函數(shù)νi=1(i=1,2,…,n)時, 由定理2即得定理1中當(dāng)1

2定理1的證明

各向異性Sobolev空間形如

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

其范數(shù)分別為

(2)

注意到

從而由式(2)可推出

下面利用H?lder不等式證明定理1.

(3)

(4)

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

(5)

且

(6)

由于φ在[0,+∞)的凸性等價于

(7)

(8)

因為

(9)

則式(8)中的第一項乘以式(9)可得

(10)

由式(10)和引理1可得對數(shù)各向異性Sobolev不等式.定理1得證.

3定理2的證明

對于每個10,令

那么帶權(quán)的各向異性空間形如

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

其范數(shù)分別為

(11)

其中u∈Lpmax(Ω)∩Lpm(Ω).現(xiàn)在使用定理1中的證明過程,就可由(11)得不等式

(12)

再由式(12)和引理2即得對數(shù)帶權(quán)各向異性Sobolev不等式.定理2得證.

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編輯、校對：師瑯

文章編號:1006-8341(2016)02-0166-05

DOI：10.13338/j.issn.1006-8341.2016.02.006

收稿日期：2015-08-23

基金項目:國家自然科學(xué)基金資助項目(11271299)

通訊作者:馮廷福(1986—),男，云南省大理白族自治州人,西北工業(yè)大學(xué)博士研究生,研究方向為偏微分方程及其應(yīng)用.E-mail:ftfml@mail.nwpu.edu.cn

中圖分類號：O 178

文獻標(biāo)識碼：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 H?lder 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:H?lder inequality;logarithmic anisotropic Sobolev inequality;logarithmic weighted anisotropic Sobolev inequality

引文格式：馮廷福,董艷.對數(shù)各向異性Sobolev不等式[J].紡織高?；A(chǔ)科學(xué)學(xué)報,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.