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,




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


The integration of this inequalityMathworldPlanetmath yields,


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:


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


By the arithmetic-geometric means inequality, one obtains


One finally concludes


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


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


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

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




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


One has, by the Hölder inequality,


Therefore, the Sobolev inequality yields


This yields


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].


  • 1 Haïm Brezis, Analyse fonctionnelle, Théorie et applications, Mathématiques appliquées pour la maîtrise, Masson, Paris, 1983.[MR85a:46001]
  • 2 Jürgen Jost, Partial Differential EquationsMathworldPlanetmath, Graduate Texts in Mathematics, Springer, 2002,[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