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 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

[math]\displaystyle{ \lim_{k \to \infty} u_{k} = u \mbox{ in } L^{1} (\Omega; \mathbf{R}^{m}) }[/math]

and

[math]\displaystyle{ \lim_{k \to \infty} \nabla u_{k} = v }[/math]

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

[math]\displaystyle{ \| (u, v) \| := \| u \|_{L^{1}} + \| v \|_{M}, }[/math]

i.e. the sum of the L¹ and total variation norms of the two components.

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. doi:10.21136/CPM.1972.117746. ISSN 0528-2195.  MR0313798