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 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 &Omega; be a bounded domain in n-dimensional Euclidean space with smooth boundary. The Souček space W1,&mu;(&Omega;; Rm) is defined to be the space of all ordered pairs (u, v), where


 * u lies in the Lebesgue space L1(&Omega;; Rm);
 * v (thought of as the gradient of u) is a regular Borel measure on the closure of &Omega;;
 * there exists a sequence of functions uk in the Sobolev space W1,1(&Omega;; Rm) 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-&lowast; in the space of all Rm&times;n-valued regular Borel measures on the closure of &Omega;.

Properties

 * The Souček space W1,&mu;(&Omega;; Rm) 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.