Souček space

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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.

[edit] 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 L¹(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

$\lim_{k \to \infty} u_{k} = u \mbox{ in } L^{1} (\Omega; \mathbf{R}^{m})$

and

$\lim_{k \to \infty} \nabla u_{k} = v$

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

[edit] 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 L¹ and total variation norms of the two components.

[edit] References

Souček, Jiří (1972). "Spaces of functions on domain Ω, whose k-th derivatives are measures defined on Ω̅". Časopis Pěst. Mat. 97: 10–46, 94. ISSN 0528-2195. MR 0313798