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单词 SobolevInequality
释义

Sobolev inequality


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

p*:=npn-p.

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 Rnwith LipschitzPlanetmathPlanetmath boundary.Then there is a constant C>0 such that, for all uW1,p(Ω) one has

uLp*(Ω)CuLp(Ω).

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.

TitleSobolev inequality
Canonical nameSobolevInequality
Date of creation2013-03-22 15:05:14
Last modified on2013-03-22 15:05:14
Ownerpaolini (1187)
Last modified bypaolini (1187)
Numerical id10
Authorpaolini (1187)
Entry typeTheorem
Classificationmsc 46E35
SynonymSobolev embedding
Synonymsobolev immersion
SynonymGagliardo Nirenberg inequality
Related topicLpSpace
DefinesSobolev conjugate
DefinesSobolev exponent
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更新时间:2025/5/4 15:57:09