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, 2, 3, 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, 4, 5, 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

${{\int }_{\Omega }}\frac{{{\left| u \right|}^{p}}}{\parallel u\parallel _{{{L}^{p}}}^{p}}~log\left( \frac{{{\left| u \right|}^{p}}}{\parallel u\parallel _{{{L}^{p}}}^{p}} \right)dx\le \frac{p}{n}log\left( \frac{c\parallel Du{{\parallel }^{p}}_{{{L}^{p}}}}{\parallel u{{\parallel }^{p}}_{{{L}^{p}}}} \right),$

(1)

其中u∈W₀^1,p (Ω),1≤p<+∞,c为正常数.受此启发,本文给出各向异性Sobolev不等式^{[8, 9, 10, 11, 12, 13, 14]}的几个推广.即利用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^{[15, 16, 17, 18]}建立对数各向异性Sobolev不等式和对数带权各向异性Sobolev不等式,从而将式(1)从各向同性的情形推广到各向异性. 下文出现的c均表示正常数.本文总设Ω是Rⁿ中的一个有界开集.本文主要结果如下.

定理 1(对数各向异性Sobolev不等式) 设u∈W₀^1,(p_i) (Ω),其中1≤p_i<+∞(i=1,2,…,n),记$\bar{p}={{\left( \frac{1}{n}\sum\limits_{i=1}^{n}{\frac{1}{{{p}_{i}}}} \right)}^{-1}},{{\bar{p}}^{*}}=\frac{n\bar{p}}{n-\bar{p}},{{p}_{max}}=\underset{1\le i\le n}{\mathop{max{{p}_{i}}}}\,$,则当$\bar p < n$且${\bar p^*} > {p_{max}}$时,有

$\begin{align} & {{\int }_{\Omega }}\frac{{{\left| u \right|}^{{{p}_{max}}}}}{\parallel u\parallel _{_{{{L}^{{{p}_{max}}}}}}^{{{p}_{max}}}}log\left( \frac{{{\left| u \right|}^{{{p}_{max}}}}}{\parallel u\parallel _{_{{{L}^{{{p}_{max}}}}}}^{{{p}_{max}}}} \right)dx\le \frac{1}{\frac{1-{{p}_{max}}}{{{{\bar{p}}}^{*}}}}log\left( \frac{c\prod\limits_{i=1}^{n}{\parallel {{D}_{i}}u\parallel _{_{{{L}^{{{p}_{i}}}}}}^{{{p}_{ma{{x}^{/n}}}}}}}{\parallel u\parallel _{_{{{L}^{{{p}_{max}}}}}}^{{{p}_{max}}}} \right) \\ & ~ \\ \end{align}$

成立.并可得

${{\int }_{\Omega }}\frac{{{\left| u \right|}^{{{p}_{max}}}}}{\parallel u\parallel _{_{{{L}^{{{p}_{max}}}}}}^{{{p}_{max}}}}log\left( \frac{{{\left| u \right|}^{{{p}_{max}}}}}{\parallel u\parallel _{_{{{L}^{{{p}_{max}}}}}}^{{{p}_{max}}}} \right)dx\le \frac{1}{1-\frac{{{p}_{max}}}{{{{\bar{p}}}^{*}}}}log\left( \frac{{{\left( \frac{c}{n}\sum\limits_{i=1}^{n}{\parallel {{D}_{i}}u{{\parallel }_{{{L}^{{{p}_{i}}}}}}} \right)}^{{{p}_{^{max}}}}}}{\parallel u\parallel _{_{{{L}^{{{p}_{max}}}}}}^{{{p}_{max}}}} \right),$

注 若 1≤p_i=p_max=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.

定理 2(对数带权各向异性Sobolev不等式) 设u∈W₀^1,(p_i) (Ω,ν_i),其中1<p_ii是非负的且满足$\frac{1}{{{\nu }_{i}}}\in {{L}^{{{m}_{i}}}}$(Ω),m_i>0,i=1,2,…,n.设${{m}_{i}}\ge \frac{1}{{{p}_{i}}-1}$,并记${{p}_{m}}=\frac{1}{\sum\limits_{i=1}^{n}{\frac{{{m}_{i}}}{{{m}_{i}}{{p}_{i}}}}}$,则当p_m>p_max时,有

${{\int }_{\Omega }}\frac{{{\left| u \right|}^{{{p}_{max}}}}}{\parallel u\parallel _{{{L}^{{{p}_{\max }}}}}^{{{p}_{max}}}}~log\left( \frac{{{\left| u \right|}^{{{p}_{max}}}}}{\parallel u\parallel _{{{L}^{{{p}_{\max }}}}}^{{{p}_{max}}}} \right)dx\le \frac{1}{1-\frac{{{p}_{max}}}{{{p}_{m}}}}log\left( \frac{c{{\left( \prod\limits_{i=1}^{n}{{{\left( {{\int }_{\Omega }}{{\nu }_{i}}|{{D}_{i}}u{{|}^{{{p}_{i}}}}dx \right)}^{1/{{p}_{i}}}}} \right)}^{^{{{p}_{max}}/n}}}}{\parallel u\parallel _{{{L}^{{{p}_{\max }}}}}^{{{p}_{max}}}} \right).$

进一步有

${{\int }_{\Omega }}\frac{{{\left| u \right|}^{{{p}_{max}}}}}{\parallel u\parallel _{{{L}^{{{p}_{\max }}}}}^{{{p}_{max}}}}~log\left( \frac{{{\left| u \right|}^{{{p}_{max}}}}}{\parallel u\parallel _{{{L}^{{{p}_{\max }}}}}^{{{p}_{max}}}} \right)dx\le \frac{1}{1-\frac{{{p}_{max}}}{{{p}_{m}}}}log\left( \frac{{{\left( \frac{c}{n}\sum\limits_{i=1}^{n}{{{\left( {{\int }_{\Omega }}{{\nu }_{i}}|{{D}_{i}}u{{|}^{{{p}_{i}}}}dx \right)}^{1/{{p}_{i}}}}} \right)}^{^{{{p}_{max}}}}}}{\parallel u\parallel _{{{L}^{{{p}_{\max }}}}}^{{{p}_{max}}}} \right).$

注 当权函数ν_i=1(i=1,2,…,n)时,由定理2即得定理1中当1<p_i 2 定理1的证明

设Ω是Rⁿ中的一个有界开集,对于1≤p_i<+∞(i=1,2,…,n),用p_max表示p_i中的最大值,p_min表示p_i中的最小值,表示p_i的调和均值,分别记为

${{p}_{max}}=\underset{1\le i\le n}{\mathop{max{{p}_{i}}}}\,,{{p}_{min}}=\underset{1\le i\le n}{\mathop{min{{p}_{i}}}}\,,=\bar{p}={{\left( \frac{1}{n}\sum\limits_{i=1}^{n}{\frac{1}{{{p}_{i}}}} \right)}^{-1}}.$

各向异性Sobolev空间形如

$\begin{align} & {{W}^{1,({{p}_{i}})}}\left( \Omega \right)=\{u\in {{W}^{1,1}}\left( \Omega \right):{{D}_{i}}u\in {{L}^{{{p}_{i}}}}\left( \Omega \right),i=1,\ldots ,n\}, \\ & {{W}^{1,({{p}_{i}})}}_{0}\left( \Omega \right)=\{u\in W_{0}^{1,1}~\left( \Omega \right):{{D}_{i}}u\in {{L}^{{{p}_{i}}}}\left( \Omega \right),i=1,\ldots ,n\}, \\ \end{align}$

其范数分别为

$\begin{align} & \parallel u{{\parallel }_{W_{0}^{^{1,({{p}_{i}})}}(\Omega )}}={{\int }_{\Omega }}\left| u \right|dx+{{\sum\limits_{i=1}^{n}{\left( {{\int }_{\Omega }}|{{D}_{i}}u{{|}^{{{p}_{i}}}}dx \right)}}^{1/{{p}_{i}}}}, \\ & \parallel u{{\parallel }_{W_{0}^{^{1,({{p}_{i}})}}(\Omega )}}={{\sum\limits_{i=1}^{n}{\left( {{\int }_{\Omega }}|{{D}_{i}}u{{|}^{{{p}_{i}}}}dx \right)}}^{1/{{p}_{i}}}}. \\ \end{align}$

引理 1(各向异性Sobolev不等式) 若$\sum\limits_{i=1}^{n}{\frac{1}{{{p}^{i}}}}>1$,即$\bar p < n$,则存在一个正常数c,使得$\forall u\in W_{0}^{^{1,({{p}_{i}})}}$(Ω),有

${{\left( {{\int }_{\Omega }}{{\left| u \right|}^{{{^{{\bar{p}}}}^{*}}}}dx \right)}^{1{{/}^{{\bar{p}}}}^{*}}}\le c{{\left( \prod\limits_{i=1}^{n}{{{\left( {{\int }_{\Omega }}|{{D}_{i}}u{{|}^{{{p}_{i}}}}dx \right)}^{1/{{p}_{i}}}}} \right)}^{1/n}}=c\prod\limits_{i=1}^{n}{\parallel {{D}_{i}}u\parallel _{_{{{L}^{{{p}_{i}}}}(\Omega )}}^{^{1/n}}},$

(2)

其中$\bar{p}*$满足$\frac{1}{{{^{{\bar{p}}}}^{*}}}=\frac{1}{^{{\bar{p}}}}-\frac{1}{n}$.

注意到

$\prod\limits_{i=1}^{n}{\parallel {{D}_{i}}u\parallel _{_{{{L}^{{{p}_{i}}}}(\Omega )}}^{^{1/n}}}\le \frac{\sum\limits_{i=1}^{n}{\parallel {{D}_{i}}u{{\parallel }_{{{L}^{{{p}_{i}}}}(\Omega )}}}}{n},$

从而由式(2)可推出

${{\left( {{\int }_{\Omega }}{{\left| u \right|}^{{{^{{\bar{p}}}}^{*}}}}dx \right)}^{1{{/}^{{\bar{p}}}}^{*}}}\le \frac{c}{n}\sum\limits_{i=1}^{n}{\parallel {{D}_{i}}u{{\parallel }_{{{L}^{{{p}_{i}}}}(\Omega )}}}.$

下面利用Hölder不等式证明定理 1.

定理 1的证明 ∀θ∈[0,1],记r满足$\frac{1}{r}=\frac{\theta }{{{p}_{\max }}}+\frac{1-\theta }{{{^{{\bar{p}}}}^{*}}}$,则p_max≤r≤$\bar{p}*$,从而由Hölder不等式可得

$\parallel u{{\parallel }_{{{L}^{r}}}}\le \parallel u\parallel _{{{L}^{{{p}_{_{\max }}}}}}^{\theta }\parallel u\parallel _{{{L}^{{{^{{\bar{p}}}}^{*}}}}}^{1-\theta },$

(3)

其中$u\in {{L}^{{{p}_{max}}}}\left( \Omega \right)\cap \text{ }{{L}^{\bar{p}*}}$(Ω).对式(3)两边取对数得

$log(\parallel u{{\parallel }_{{{L}^{r}}}})\le \theta log(\parallel u{{\parallel }_{{{L}^{{{p}_{max}}}}}})+\left( 1-\theta \right)log(\parallel u{{\parallel }_{{{L}^{{{^{{\bar{p}}}}^{*}}}}}}).$

(4)

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

$\varphi \left( \frac{1}{r} \right)=\varphi \left( \frac{\theta }{{{p}_{max}}}+\frac{1}{{{^{{\bar{p}}}}^{*}}} \right)\le \theta \varphi \left( \frac{1}{{{p}_{max}}} \right)+\left( 1-\theta \right)\varphi \left( \frac{1}{{{^{{\bar{p}}}}^{*}}} \right).$

这里$\varphi \left( \frac{1}{r} \right)=log(\parallel u{{\parallel }_{{{L}^{r}}}}),\varphi \left( \frac{1}{{{p}_{max}}} \right)=log(\parallel u{{\parallel }_{{{L}^{{{p}_{max}}}}}}),\varphi \left( \frac{1}{{{^{{\bar{p}}}}^{*}}} \right)=log(\parallel u{{\parallel }_{{{L}^{{{^{{\bar{p}}}}^{*}}}}}})$.令$\frac{1}{r}=h$,则

$\varphi \left( h \right)=\left( hlog{{\int }_{\Omega }}{{\left| u \right|}^{1/h}}dx \right),$

(5)

且

$\varphi \prime \left( h \right)=log\left( {{\int }_{\Omega }}{{\left| u \right|}^{\frac{1}{h}}}dx \right)-\frac{1}{h}\frac{{{\int }_{\Omega }}log({{\left| u\left| ) \right|u \right|}^{1/h}}dx}{{{\int }_{\Omega }}{{\left| u \right|}^{1/h}}dx}.~$

(6)

由于φ在［0,+∞)的凸性等价于

$\varphi \prime \left( h \right)\ge \frac{\varphi ({{h}_{0}})-\varphi \left( h \right)}{{{h}_{0}}-h},$

(7)

这里h>h₀≥0,则取${{h}_{0}}=\frac{1}{{{^{{\bar{p}}}}^{*}}},h=\frac{1}{{{p}_{max}}}$,把式(5),(6)代入式(7)中得

$-log\left( {{\int }_{\Omega }}{{\left| u \right|}^{{{p}_{max}}}}dx \right)+\frac{{{\int }_{\Omega }}log({{\left| u \right|}^{{{p}_{max}}}}){{\left| u \right|}^{{{p}_{max}}}}dx}{{{\int }_{\Omega }}{{\left| u \right|}^{{{p}_{max}}}}dx}\le \frac{log{{\left( {{\int }_{\Omega }}{{\left| u \right|}^{{{^{{\bar{p}}}}^{*}}}}dx \right)}^{{{p}_{max}}{{/}^{{\bar{p}}}}^{*}}}-log\left( {{\int }_{\Omega }}{{\left| u \right|}^{{{p}_{max}}}}dx \right)}{1-\frac{{{p}_{max}}}{{{^{{\bar{p}}}}^{*}}}}.$

(8)

因为

${{\int }_{\Omega }}\frac{{{\left| u \right|}^{{{p}_{max}}}}}{\parallel u\parallel _{{{L}^{{{p}_{max}}}}}^{{{p}_{max}}}}dx=1,$

(9)

则式(8)中的第一项乘以式(9)可得

${{\int }_{\Omega }}\frac{{{\left| u \right|}^{{{p}_{max}}}}}{\parallel u\parallel _{{{L}^{{{p}_{max}}}}}^{{{p}_{max}}}}~log\left( \frac{{{\left| u \right|}^{{{p}_{max}}}}}{\parallel u\parallel _{{{L}^{{{p}_{max}}}}}^{{{p}_{max}}}} \right)dx\le \frac{1}{1-{{p}_{max}}{{/}^{{\bar{p}}}}^{*}}log\left( \frac{\parallel u\parallel _{{{L}^{{{^{{\bar{p}}}}^{*}}}}}^{{{p}_{max}}}}{\parallel u\parallel _{{{L}^{{{p}_{max}}}}}^{{{p}_{max}}}} \right).$

(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1<p_i<n(i=1,2,…,n)和每个m=(m₁,…,m_n)∈Rⁿ,设m_i>0,令

${{p}_{m}}=n/\left( \sum\limits_{i=1}^{n}{\frac{1+{{m}_{i}}}{{{m}_{i}}{{p}_{i}}}-1} \right),$

则当${{m}_{i}}\ge \frac{1}{{{p}_{i}}-1}$时,由

$\sum\limits_{i=1}^{n}{\frac{1+{{m}_{i}}}{{{m}_{i}}{{p}_{i}}}}=\sum\limits_{i=1}^{n}{\left( \frac{1}{{{m}_{i}}}+1 \right)\frac{1}{{{p}_{i}}}\le }\sum\limits_{i=1}^{n}{\left( {{p}_{i}}-1+1 \right)}\frac{1}{{{p}_{i}}}=n,$

可得${{p}_{m}}\ge \frac{n}{n-1}>1$.设权函数ν_i是Ω上的非负可积函数且满足

${{\nu }_{i}}\in L_{_{loc}}^{1}~\left( \Omega \right),{{\left( \frac{1}{{{\nu }_{i}}} \right)}^{{{m}_{i}}}}\in {{L}^{1}}\left( \Omega \right),{{m}_{i}}\ge \frac{1}{{{p}_{i}}-1}.$

那么带权的各向异性空间形如

$\begin{align} & {{W}^{1,({{p}_{i}})}}(\Omega ,{{\nu }_{i}})=\{u\in {{W}^{1,1}}\left( \Omega \right):{{\nu }_{i}}|{{D}_{i}}u{{|}^{{{p}_{i}}}}\in {{L}^{1}}\left( \Omega \right),i=1,\ldots ,n\}, \\ & W_{_{0}}^{^{1,({{p}_{i}})}}(\Omega ,{{\nu }_{i}})=\{u\in W_{0}^{1,1}\left( \Omega \right):{{\nu }_{i}}|{{D}_{i}}u{{|}^{{{p}_{i}}}}\in {{L}^{1}}\left( \Omega \right),i=1,\ldots ,n\}. \\ \end{align}$

其范数分别为

$\begin{align} & \parallel u{{\parallel }_{{{W}^{1,({{p}_{i}})}}(\Omega ,{{\nu }_{i}})}}={{\int }_{\Omega }}\left| u \right|dx+{{\sum\limits_{i=1}^{n}{\left( {{\int }_{\Omega }}{{\nu }_{i}}|{{D}_{i}}u{{|}^{{{p}_{i}}}}dx \right)}}^{1/{{p}_{i}}}}, \\ & \parallel u{{\parallel }_{W_{0}^{^{1,({{p}_{i}})}}\left( \Omega ,{{\nu }_{i}} \right)}}={{\sum\limits_{i=1}^{n}{\left( {{\int }_{\Omega }}{{\nu }_{i}}|{{D}_{i}}u{{|}^{{{p}_{i}}}}dx \right)}}^{1/{{p}_{i}}}}. \\ \end{align}$

引理 2(带权的各向异性Sobolev不等式) ∀i∈｛1,2,…,n｝,设${{m}_{i}}\ge \frac{1}{{{p}_{i}}-1}$和$\frac{1}{{{\nu }_{i}}}\in {{L}^{{{m}_{i}}}}$(Ω),则$W_{0}^{1,({{p}_{i}})}(\Omega ,{{\nu }_{i}})\subset {{L}^{{{p}_{m}}}}\left( \Omega \right)$,即存在一个正常数c,使得∀u∈W₀^1,(p_i) (Ω,ν_i),有

$\begin{align} & {{\left( {{\int }_{\Omega }}\left| u \right|{{p}_{m}}dx \right)}^{1/{{p}_{m}}}}\le c{{\left( {{\prod\limits_{i=1}^{n}{\left( {{\int }_{\Omega }}{{\nu }_{i}}|{{D}_{i}}u{{|}^{{{p}_{i}}}}dx \right)}}^{1/{{p}_{i}}}} \right)}^{1/n}}, \\ & {{\left( {{\int }_{\Omega }}\left| u \right|{{p}_{m}}dx \right)}^{1/{{p}_{m}}}}\le \frac{c}{n}{{\sum\limits_{i=1}^{n}{\left( {{\int }_{\Omega }}{{\nu }_{i}}|{{D}_{i}}u{{|}^{{{p}_{i}}}}dx \right)}}^{1/{{p}_{i}}}}. \\ \end{align}$

定理 2的证明 ∀θ∈［0,1],记r满足$\frac{1}{r}=\frac{\theta }{{{p}_{max}}}+\frac{1-\theta }{{{p}_{m}}}$,则p_max≤r≤p_m,从而有Hölder不等式

$\parallel u{{\parallel }_{{{L}^{r}}}}\le \parallel u\parallel _{_{{{L}^{{{p}_{max}}}}}}^{\theta }\|u\|_{_{{{L}^{{{p}_{m}}}}}}^{1-\theta },$

(11)

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

${{\int }_{\Omega }}\frac{{{\left| u \right|}^{{{p}_{max}}}}}{\parallel u\parallel _{{{L}^{{{p}_{max}}}}}^{{{p}_{max}}}}~log\left( \frac{{{\left| u \right|}^{{{p}_{max}}}}}{\parallel u\parallel _{{{L}^{{{p}_{max}}}}}^{{{p}_{max}}}} \right)dx\le \frac{1}{1-{{p}_{max}}/{{p}_{m}}}log\left( \frac{\parallel u\parallel _{{{L}^{{{p}_{m}}}}}^{{{p}_{max}}}}{\parallel u\parallel _{{{L}^{{{p}_{max}}}}}^{{{p}_{max}}}} \right).$

(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.