Sobolev inequality

For 1p<n, define the Sobolev conjugate of p as


Note that -n/p*=1-n/p.

In the following statement represent the weak derivative and W1,p(Ω) is the Sobolev spaceMathworldPlanetmath of functions uLp(Ω) whose weak derivative u is itself in Lp(Ω).

Theorem 1

Assume that p[1,n) and let Ω be a boundedPlanetmathPlanetmathPlanetmathPlanetmath, open subset of Rn with LipschitzPlanetmathPlanetmath boundary. Then there is a constant C>0 such that, for all uW1,p(Ω) one has


We can restate the previous Theorem by saying that the Sobolev space W1,p(Ω) is a subspaceMathworldPlanetmathPlanetmath of the Lebesgue space Lp*(Ω) and that the inclusion mapMathworldPlanetmath i:W1,p(Ω)Lq*(Ω) is continuousMathworldPlanetmathPlanetmath.

Title Sobolev inequality
Canonical name SobolevInequality
Date of creation 2013-03-22 15:05:14
Last modified on 2013-03-22 15:05:14
Owner paolini (1187)
Last modified by paolini (1187)
Numerical id 10
Author paolini (1187)
Entry type Theorem
Classification msc 46E35
Synonym Sobolev embedding
Synonym sobolev immersion
Synonym Gagliardo Nirenberg inequality
Related topic LpSpace
Defines Sobolev conjugate
Defines Sobolev exponent