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 (spaces of smooth functions defined on the boundary). Commonly considered types of domains are domains with continuous boundary, Lipschitz boundary, C¹ boundary, and so forth.

Bounded domain is a domain which is a bounded set. Exterior, or external, domain is 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.

See also

• Analytic polyhedron
• Caccioppoli set
• Hartogs domain
• Lipschitz domain