proof of Sobolev inequality for
Step 1: is smooth and
First suppose is a compactly supported smooth function, and let denote a basis of . For every ,
Note that does not depend on . One also has
The integration of this inequality yields,
Since does not depend on , we can apply the generalized Hölder inequality with for the integration with respect to in order to obtain:
The repetition of this process for the variables gives
By the arithmetic-geometric means inequality, one obtains
One finally concludes
Step 2: general and
Step 3: and is smooth
Suppose and is a smooth compactly supported function. Let
Since is smooth, (It is however not necessarily smooth), and its weak derivative is
One has, by the Hölder inequality,
Therefore, the Sobolev inequality yields
Step 4: and
This is done as step 2.
- 1 HaÃÂ¯m Brezis, Analyse fonctionnelle, ThÃÂ©orie et applications, MathÃÂ©matiques appliquÃÂ©es pour la maÃÂ®trise, Masson, Paris, 1983. http://www.ams.org/mathscinet-getitem?mr=0697382[MR85a:46001]
- 2 JÃÂ¼rgen Jost, Partial Differential Equations, Graduate Texts in Mathematics, Springer, 2002, http://www.ams.org/mathscinet-getitem?mr=1919991[MR:2003f:35002].
- 3 Michel Willem, Analyse fonctinnelle ÃÂ©lÃÂ©mentaire, Cassini, Paris, 2003.
|Title||proof of Sobolev inequality for|
|Date of creation||2013-03-22 15:05:22|
|Last modified on||2013-03-22 15:05:22|
|Last modified by||vanschaf (8061)|