Domain (mathematical analysis)

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, C¹ 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 ℂⁿ.

Historical notes

Definition. Eine offene Punktmenge heißt zusammenhängend, wenn man sie nicht als Summe von zwei offenen Punktmengen darstellen kann. Eine offene zusammenhängende Punktmenge heißt ein Gebiet.^[1]

— Constantin Carathéodory, (Carathéodory 1918, p. 222)

According to Hans Hahn,^[2] the concept of a domain as an open connected set was introduced by Constantin Carathéodory in his famous book (Carathéodory 1918). Hahn also remarks that the word "Gebiet" ("Domain") was occasionally previously used as a synonym of open set.^[3]

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,^[4][5] and reserves the term "domain" to identify an internally connected,^[6] perfect set, each point of which is an accumulation point of interior points,^[4] following his former master Mauro Picone:^[7] according to this convention, if a set A is a region then its closure A is a domain.^[4]

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 ISBN 0-8176-4097-5.
• .
• , translated from the Italian by Zane C. Motteler.
• (Review of the whole volume I) (available from the "Edizione Nazionale Mathematica Italiana").