In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, Souček spaces are generalizations of
Sobolev spaces
In mathematics, a Sobolev space is a vector space of functions equipped with a normed space, norm that is a combination of Lp norm, ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a ...
, named after the
Czech
Czech may refer to:
* Anything from or related to the Czech Republic, a country in Europe
** Czech language
** Czechs, the people of the area
** Czech culture
** Czech cuisine
* One of three mythical brothers, Lech, Czech, and Rus
*Czech (surnam ...
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
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
In the area of mathematics known as functional analysis, a reflexive space is a locally convex topological vector space for which the canonical evaluation map from X into its bidual (which is the strong dual of the strong dual of X) is a homeomo ...
; since ''W''
1,1 is not reflexive, it is not always true that a bounded sequence has a
weakly convergent subsequence
In mathematics, a subsequence of a given sequence is a sequence that can be derived from the given sequence by deleting some or no elements without changing the order of the remaining elements. For example, the sequence \langle A,B,D \rangle is a ...
, which is a desideratum in many applications.
Definition
Let Ω be a
bounded domain
In mathematical analysis, a domain or region is a non-empty, connected, and open set in a topological space. In particular, it is any non-empty connected open subset of the real coordinate space or the complex coordinate space . A connected open ...
in ''n''-dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
with smooth
boundary. The Souček space ''W''
1,''μ''(Ω; R
''m'') is defined to be the space of all
ordered pair
In mathematics, an ordered pair, denoted (''a'', ''b''), is a pair of objects in which their order is significant. The ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a''), unless ''a'' = ''b''. In contrast, the '' unord ...
s (''u'', ''v''), where
* ''u'' lies in the
Lebesgue space ''L''
1(Ω; R
''m'');
* ''v'' (thought of as the gradient of ''u'') is a
regular
Regular may refer to:
Arts, entertainment, and media Music
* "Regular" (Badfinger song)
* Regular tunings of stringed instruments, tunings with equal intervals between the paired notes of successive open strings
Other uses
* Regular character, ...
Borel measure
In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require additional restrictions on the measure, as described below.
...
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
::
:and
::
:
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
In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...
when equipped with the
norm
Norm, the Norm or NORM may refer to:
In academic disciplines
* Normativity, phenomenon of designating things as good or bad
* Norm (geology), an estimate of the idealised mineral content of a rock
* Norm (philosophy), a standard in normative e ...
given by
::
:i.e. the sum of the ''L''
1 and
total variation
In mathematics, the total variation identifies several slightly different concepts, related to the (local property, local or global) structure of the codomain of a Function (mathematics), function or a measure (mathematics), measure. For a real ...
norms of the two components.
References
*
{{Functional analysis
Banach spaces
Sobolev spaces