Sobolev space

We define the Sobolev spacesMathworldPlanetmath of functions Wm,p⁒(Ξ©) where Ξ© is an open subset of 𝐑n, mβ‰₯0 is an integer and p∈[1,+∞].

The spaces W0,p⁒(Ξ©) are simply defined to be the spaces Lp⁒(Ξ©) of Lebesgue p-summable functions. We then define the space Wm,p⁒(Ξ©) to be the space of functions u∈Lp⁒(Ξ©) which have weak derivatives g=(g1,…,gn) such that gi∈Wm-1,p⁒(Ξ©).

The space Wm,p turns out to be a Banach spaceMathworldPlanetmath when endowed with the norm


i.e.Β the sum of the Lp norms of u and of all weak derivatives of u up to the m-th order.

Of particular interest are the spaces Hm⁒(Ω):=Wm,2⁒(Ω) which turn out to be Hilbert spacesMathworldPlanetmath with the scalar productMathworldPlanetmath given by

Title Sobolev space
Canonical name SobolevSpace
Date of creation 2013-03-22 14:54:55
Last modified on 2013-03-22 14:54:55
Owner paolini (1187)
Last modified by paolini (1187)
Numerical id 10
Author paolini (1187)
Entry type Definition
Classification msc 46E35
Synonym Sobolev function
Related topic WeakDerivative