proof of Sobolev inequality for Ω=𝐑n


Step 1: u is smooth and p=1

First suppose u is a compactly supported smooth functionMathworldPlanetmath, and let (ek)1≤k≤n denote a basis of 𝐑n. For every 1≤k≤n,

u⁢(x)=∫-∞0∂⁡u∂⁡xk⁢(x+s⁢ek)⁢𝑑s.

Therefore,

|u⁢(x)|≤Sk⁢(x):=∫𝐑|∂⁡u∂⁡xk⁢(x1,…,xk-1,s,xk+1,…,xn)|⁢𝑑s.

Note that Sk does not depend on xk. One also has

|u⁢(x)|n/(n-1)≤∏k=1n|Sk⁢(x)|1/(n-1).

The integration of this inequalityMathworldPlanetmath yields,

∫𝐑n|u⁢(x)|n/(n-1)⁢𝑑x≤∫𝐑n∏k=1n|Sk⁢(x)|1/(n-1)⁢d⁢x.

Since S1 does not depend on xk, we can apply the generalized Hölder inequality with n-1 for the integration with respect to x1 in order to obtain:

∫𝐑n|u⁢(x)|n/(n-1)⁢𝑑x≤∫𝐑n-1S1⁢(x)⁢∏k=2n(∫𝐑Sk⁢(x)⁢𝑑x1)1/(n-1)⁢d⁢x1⁢…⁢d⁢xn.

The repetition of this process for the variables x2,…,xn gives

∫𝐑n|u⁢(x)|n/(n-1)⁢𝑑x≤∏k=1n(∫𝐑n|∂⁡u∂⁡xk|⁢𝑑x)1/(n-1).

By the arithmetic-geometric means inequality, one obtains

∫𝐑n|u⁢(x)|n/(n-1)⁢𝑑x≤n-n/(n-1)⁢(∑k=1n(∫𝐑n|∂⁡u∂⁡xk|⁢𝑑x))n/(n-1).

One finally concludes

∥u∥Ln/(n-1)≤n1/2-n/(n-1)⁢∥∇⁡u∥Ln/(n-1).

Step 2: general u and p=1

In general if u∈W1,1⁢(𝐑n). It can be approximated by a sequence of compactly supported smooth functions (um). By step 1, one has

∥um-uℓ∥Ln/(n-1)≤n1/2-n/(n-1)⁢∥∇⁡um-∇⁡uℓ∥L1.

therefore (um) is a Cauchy sequenceMathworldPlanetmath in Ln/(n-1)⁢(𝐑n). Since it converges to u in L1⁢(𝐑n), the limit of (um) is u in Ln/(n-1)⁢(𝐑n) and one has

∥u∥Ln/(n-1)≤n1/2-n/(n-1)⁢∥∇⁡u∥Ln/(n-1).

Step 3: 1<p<n and u is smooth

Suppose 1<p<n and u is a smooth compactly supported function. Let

r=p⁢(n-1)n-p

and

v=|u|r.

Since u is smooth, v∈W1,1 (It is however not necessarily smooth), and its weak derivative is

∇⁡v=r⁢u⁢|u|r-2⁢∇⁡u.

One has, by the Hölder inequality,

∥∇v∥L1⁢(𝐑N)≤r∥|u|r∥Lp/p-1⁢(𝐑N)∥∇u∥Lp⁢(𝐑N)=r∥u∥Ln⁢p/(n-p)⁢(𝐑N)r-1∥∇u∥Lp⁢(𝐑N)

Therefore, the Sobolev inequality yields

∥u∥Ln⁢p/(n-p)⁢(𝐑N)r=∥v∥Ln/(n-1)⁢(𝐑N)≤rn1/2-n/(n-1)∥u∥Ln⁢p/(n-p)⁢(𝐑N)r-1∥∇u∥Lp⁢(𝐑N).

This yields

∥u∥Ln⁢p/(n-p)⁢(𝐑N)≤n1/2-n/(n-1)p⁢(n-1)n-p∥∇u∥Lp⁢(𝐑N).

Step 4: 1<p<n and u∈W1,p

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].

References

  • 1 Haïm Brezis, Analyse fonctionnelle, Théorie et applications, Mathématiques appliquées pour la maîtrise, Masson, Paris, 1983. http://www.ams.org/mathscinet-getitem?mr=0697382[MR85a:46001]
  • 2 Jürgen Jost, Partial Differential EquationsMathworldPlanetmath, Graduate Texts in Mathematics, Springer, 2002, http://www.ams.org/mathscinet-getitem?mr=1919991[MR:2003f:35002].
  • 3 Michel Willem, Analyse fonctinnelle élémentaire, Cassini, Paris, 2003.
Title proof of Sobolev inequality for Ω=𝐑n
Canonical name ProofOfSobolevInequalityForOmegamathbfRn
Date of creation 2013-03-22 15:05:22
Last modified on 2013-03-22 15:05:22
Owner vanschaf (8061)
Last modified by vanschaf (8061)
Numerical id 14
Author vanschaf (8061)
Entry type Proof
Classification msc 46E35