Thermodynamic integration

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

$\underset{k\to \mathrm{\infty }}{\lim }{u}_k=u\text{ in }{L}^1(\mathrm{\Omega };{\mathbf{R}}^m)$

and

$\underset{k\to \mathrm{\infty }}{\lim }\mathrm{\nabla }{u}_k=v$

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

$\Vert (u,v)\Vert :=\Vert u{\Vert }_{L^1}+\Vert v{\Vert }_M,$

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

References

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