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 W1,1 is not a reflexive space; since W1,1 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 W1,μ(Ω; Rm) is defined to be the space of all ordered pairs (u, v), where
- u lies in the Lebesgue space L1(Ω; Rm);
- v (thought of as the gradient of u) is a regular Borel measure on the closure of Ω;
- there exists a sequence of functions uk in the Sobolev space W1,1(Ω; Rm) 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 Rm×n-valued regular Borel measures on the closure of Ω.
Properties
- The Souček space W1,μ(Ω; Rm) 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
![]() | Original source: https://en.wikipedia.org/wiki/Souček space.
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