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Souček space

From Wikipedia, the free encyclopedia

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

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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 (uv), 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
and
weakly-∗ in the space of all Rm×n-valued regular Borel measures on the closure of Ω.

Properties

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  • The Souček space W1,μ(Ω; Rm) is a Banach space when equipped with the norm given by
i.e. the sum of the L1 and total variation norms of the two components.

References

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  • 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