|
|
|
|
![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
proof of Sobolev inequality for 
|
(Proof)
|
|
|
First suppose $u$ is a compactly supported smooth function, and let $(e_k)_{1 \le k \le n}$ denote a basis of $\mathbf{R}^n$ . For every $1 \le k \le n$ , $$ u(x)=\int_{-\infty}^0 \frac{\partial u}{\partial x_k}(x+se_k)\,ds. $$ Therefore, $$ \lvert u(x)\rvert \le S_k(x):=\int_{\mathbf{R}} \bigl\lvert\frac{\partial u}{\partial x_k}(x_1,\dots,x_{k-1},s,x_{k+1},\dots,x_n)\bigr\rvert\,ds. $$ Note that $S_k$ does not depend on $x_k$ . One also has $$ \lvert u(x)\rvert^{n/(n-1)}\le \prod_{k=1}^n \lvert S_k(x)\rvert^{1/(n-1)}. $$ The integration of this inequality yields, $$ \int_{\mathbf{R}^n} \vert u(x)\vert^{n/(n-1)}\,dx\le \int_{\mathbf{R}^n}\prod_{k=1}^n \lvert S_k(x)\rvert^{1/(n-1)}\,dx. $$ Since $S_1$ does not depend on $x_k$ , we can apply the generalized Hölder inequality with $n-1$ for the integration with respect to $x_1$ in order to obtain: $$ \int_{\mathbf{R}^n} \lvert u(x)\rvert^{n/(n-1)}\,dx\le \int_{\mathbf{R}^{n-1}} S_1(x)\prod_{k=2}^n \Bigl(\int_{\mathbf{R}} S_k(x)\,dx_1\Bigr)^{1/(n-1)}\,dx_1\dots dx_n. $$ The repetition of this process for the variables $x_2,\dots,x_n$ gives $$ \int_{\mathbf{R}^n} \lvert u(x)\rvert^{n/(n-1)}\,dx\le \prod_{k=1}^n \Bigl(\int_{\mathbf{R}^n} \bigl\lvert \frac{\partial u}{\partial x_k}\bigr\rvert\,dx\Bigr)^{1/(n-1)}. $$ By the arithmetic-geometric means inequality, one obtains $$ \int_{\mathbf{R}^n} \lvert u(x)\rvert^{n/(n-1)}\,dx\le n^ {-n/(n-1)}\Bigl(\sum_{k=1}^n \Bigl(\int_{\mathbf{R}^n} \bigl\lvert \frac{\partial u}{\partial x_k}\bigr\rvert\,dx\Bigr)\Bigr)^{n/(n-1)}. $$ One finally concludes $$ \lVert u \rVert_{L^{n/(n-1)}} \le n^{1/2-n/(n-1)} \lVert \nabla u \rVert_{L^{n/(n-1)}}. $$
In general if $u \in W^{1,1}(\mathbf{R}^n)$ . It can be approximated by a sequence of compactly supported smooth functions $(u_m)$ . By step 1, one has $$ \lVert u_m-u_\ell \rVert_{L^{n/(n-1)}} \le n^{1/2-n/(n-1)} \lVert \nabla u_m-\nabla u_\ell \rVert_{L^{1}}. $$ therefore $(u_m)$ is a Cauchy sequence in $L^{n/(n-1)}(\mathbf{R}^n)$ . Since it converges to $u$ in $L^{1}(\mathbf{R}^n)$ , the limit of $(u_m)$ is $u$ in $L^{n/(n-1)}(\mathbf{R}^n)$ and one has $$ \lVert u \rVert_{L^{n/(n-1)}} \le n^{1/2-n/(n-1)} \lVert \nabla u \rVert_{L^{n/(n-1)}}. $$
Suppose $1 <p <n $ and $u$ is a smooth compactly supported function. Let $$ r=\frac{p(n-1)}{n-p} $$ and $$ v=\lvert u \rvert^r. $$ Since $u$ is smooth, $v \in W^{1,1}$ (It is however not necessarily smooth), and its weak derivative is $$ \nabla v=r u \lvert u \rvert^{r-2} \nabla u. $$ One has, by the Hölder inequality, $$ \rVert \nabla v \lVert_{L^1(\mathbf{R}^N)}\le r
\rVert \lvert u\rvert^r \lVert_{L^{p/p-1}(\mathbf{R}^N)}\rVert \nabla u \lVert_{L^{p}(\mathbf{R}^N)} = r \rVert u \lVert_{L^{np/(n-p)}(\mathbf{R}^N)}^{r-1}\rVert \nabla u \lVert_{L^{p}(\mathbf{R}^N)} $$ Therefore, the Sobolev inequality yields $$ \rVert u \lVert_{L^{np/(n-p)}(\mathbf{R}^N)}^r = \rVert v \lVert_{L^{n/(n-1)}(\mathbf{R}^N)} \le r n^{1/2-n/(n-1)}\rVert u \lVert_{L^{np/(n-p)}(\mathbf{R}^N)}^{r-1}\rVert \nabla u \lVert_{L^{p}(\mathbf{R}^N)}. $$ This yields $$ \rVert u \lVert_{L^{np/(n-p)}(\mathbf{R}^N)} \le n^{1/2-n/(n-1)}\frac{p(n-1)}{n-p}\rVert \nabla u \lVert_{L^{p}(\mathbf{R}^N)}. $$
This is done as step 2.
This proof is due to Gagliardo and Nirenberg, who were the first to prove the inequality for $p=1$ . This proof can be also found in [1,2,3].
- 1
- Haïm BREZIS, Analyse fonctionnelle, Théorie et applications, Mathématiques appliquées pour la maîtrise, Masson, Paris, 1983. [MR85a:46001]
- 2
- Jürgen JOST, Partial Differential Equations, Graduate Texts in Mathematics, Springer, 2002, [MR:2003f:35002].
- 3
- Michel WILLEM, Analyse fonctinnelle élémentaire, Cassini, Paris, 2003.
|
"proof of Sobolev inequality for  " is owned by vanschaf.
|
|
(view preamble | get metadata)
Cross-references: proof, Sobolev inequality, Hölder inequality, weak derivative, function, smooth, limit, converges, Cauchy sequence, sequence, arithmetic-geometric means inequality, variables, order, generalized Hölder inequality, inequality, basis, smooth function
This is version 11 of proof of Sobolev inequality for  , born on 2005-02-23, modified 2007-05-21.
Object id is 6816, canonical name is ProofOfSobolevInequality.
Accessed 3670 times total.
Classification:
| AMS MSC: | 46E35 (Functional analysis :: Linear function spaces and their duals :: Sobolev spaces and other spaces of ``smooth'' functions, embedding theorems, trace theorems) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
