Souček space

In mathematics, Souček spaces are generalizations of Sobolev spaces, named after the Czech mathematician Jiří Souček. One of their main advantages is that they offer a way to deal with the fact that the Sobolev space "W"^1,1 is not a reflexive space; since "W"^1,1 is not reflexive, it is not always true that a bounded sequence has a weakly convergent subsequence, which is a desideratum in many applications.

Definition

Let Ω be a bounded domain in "n"-dimensional Euclidean space with smooth boundary. The Souček space "W"^1,"μ"(Ω; R^"m") is defined to be the space of all ordered pairs ("u", "v"), where

* "u" lies in the Lebesgue space "L"¹(Ω; R^"m");
* "v" (thought of as the gradient of "u") is a regular Borel measure on the closure of Ω;
* there exists a sequence of functions "u"_"k" in the Sobolev space "W"^1,1(Ω; R^"m") such that

::$lim\_\{k\; o\; infty\}\; u\_\{k\}\; =\; u\; mbox\{\; in\; \}\; L^\{1\}\; (Omega;\; mathbf\{R\}^\{m\})$

:and

::$lim\_\{k\; o\; infty\}\; abla\; u\_\{k\}\; =\; v$

:weakly-∗ in the space of all R^{"m"×"n"}-valued regular Borel measures on the closure of Ω.

Properties

* The Souček space "W"^1,"μ"(Ω; R^"m") is a Banach space when equipped with the norm given by

::$|\; (u,\; v)\; |\; :=\; |\; u\; |\_\{L^\{1\; +\; |\; v\; |\_\{M\},$

:i.e. the sum of the "L"¹ and total variation norms of the two components.

References

* cite journal
last = Souček
first = Jiří
title = Spaces of functions on domain Ω, whose "k"-th derivatives are measures defined on Ω̅
journal = Časopis Pěst. Mat.
volume = 97
year = 1972
pages = 10–46, 94
issn = 0528-2195 MathSciNet|id=0313798