In
mathematical analysis, a domain is any
connected open subset of a
finite-dimensional vector space. This is a different concept than the
domain of a function, though it is often used for that purpose, for example in
partial differential equations and
Sobolev spaces.
Various degrees of smoothness of the boundary of the domain are required for various properties of functions defined on the domain to hold, such as integral theorems (
Green's theorem,
Stokes theorem), properties of
Sobolev spaces, and to define
measures on the boundary and spaces of
traces (generalized functions defined on the boundary). Commonly considered types of domains are domains with
continuous boundary,
Lipschitz boundary,
''C''1 boundary, and so forth.
A bounded domain is a domain which is a
bounded set, while an exterior or external domain is the
interior of the
complement of a bounded domain.
In
complex analysis, a complex domain (or simply domain) is any connected open subset of the
complex plane ℂ. For example, the entire complex plane is a domain, as is the open
unit disk, the open
upper half-plane, and so forth. Often, a complex domain serves as the
domain of definition for a
holomorphic function. In the study of
several complex variables, the definition of a domain is extended to include any connected open subset of ℂ''
n''.
Historical notes
: in this definition, Carathéodory considers obviously
non empty disjoint sets.
|sign=
Constantin Carathéodory
|source=
According to
Hans Hahn, the concept of a domain as an open connected set was introduced by
Constantin Carathéodory in his famous book . Hahn also remarks that the word "''Gebiet''" ("''Domain''") was occasionally previously used as a
synonym of
open set.
However, the term "domain" was occasionally used to identify closely related but slightly different concepts. For example, in his influential
monographs on
elliptic partial differential equations,
Carlo Miranda uses the term "region" to identify an open connected set,
[See .] and reserves the term "domain" to identify an internally connected,
perfect set, each point of which is an accumulation point of interior points,
following his former master
Mauro Picone:
[See .] according to this convention, if a set is a region then its
closure is a domain.
See also
*
Analytic polyhedron
*
Caccioppoli set
*
Hartogs domain
*
Lipschitz domain
*
Region (mathematical analysis)
Notes
References
* (the
MR review refers to the third corrected edition).
* (freely available at the
Internet Archive).
*
Steven G. Krantz &
Harold R. Parks (1999) ''The Geometry of Domains in Space'',
Birkhäuser .
*.
*, translated from the Italian by Zane C. Motteler.
*{{Citation
| last = Picone
| first = Mauro
| author-link = Mauro Picone
| title = Lezioni di analisi infinitesimale
| place =
Catania
| publisher =
Circolo matematico di Catania
| series = Volume 1
| volume = Parte Prima – La Derivazione
| year = 1923
| pages = xii+351
| language = it
| url = http://mathematica.sns.it/media/volumi/462/picone_parte_I.pdf
| jfm = 49.0172.07 (Review of the whole volume I) (available from the "
Edizione Nazionale Mathematica Italiana'").
Category:Mathematical analysis
Category:Partial differential equations
Category:Topology