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