“Coercive function”的意思、由来-开放百科全书

词条

Coercive function

释义

Coercive vector fields
Coercive operators and forms
Norm-coercive mappings
(Extended valued) coercive functions
References

In mathematics, a coercive function is a function that "grows rapidly" at the extremes of the space on which it is defined. Depending on the context

different exact definitions of this idea are in use.

Coercive vector fields

A vector field f : Rⁿ → Rⁿ is called coercive if

where "" denotes the usual dot product and denotes the usual Euclidean norm of the vector x.

A coercive vector field is in particular norm-coercive since

for

, by

Cauchy Schwarz inequality.

However a norm-coercive mapping

f : Rⁿ → Rⁿ

is not necessarily a coercive vector field. For instance

the rotation

f : R² → R², f(x) = (-x₂, x₁)

by 90° is a norm-coercive mapping which fails to be a coercive vector field since

for every .

Coercive operators and forms

A self-adjoint operator where is a real Hilbert space, is called coercive if there exists a constant such that

for all in

A bilinear form is called coercive if there exists a constant such that

for all in

It follows from the Riesz representation theorem that any symmetric (defined as: for all in ), continuous ( for all in and some constant ) and coercive bilinear form has the representation

for some self-adjoint operator which then turns out to be a coercive operator. Also, given a coercive self-adjoint operator the bilinear form defined as above is coercive.

If is a coercive operator then it is a coercive mapping (in the sense of coercivity of a vector field, where one has to replace the dot product with the more general inner product). Indeed, for big (if is bounded, then it readily follows); then replacing by we get that is a coercive operator.

One can also show that the converse holds true if is self-adjoint. The definitions of coercivity for vector fields, operators, and bilinear forms are closely related and compatible.

Norm-coercive mappings

A mapping

between two normed vector spaces

and

is called norm-coercive iff

.

More generally, a function between two topological spaces and is called coercive if for every compact subset of there exists a compact subset of such that

The composition of a bijective proper map followed by a coercive map is coercive.

(Extended valued) coercive functions

An (extended valued) function

is called coercive iff

A real valued coercive function

is, in particular, norm-coercive. However, a norm-coercive function

is not necessarily coercive.

For instance, the identity function on is norm-coercive

but not coercive.

References

{{cite book|author1=Renardy, Michael |author2=Rogers, Robert C. | title=An introduction to partial differential equations | edition=Second | publisher=Springer-Verlag | location=New York, NY | year=2004 | pages=xiv+434 | isbn=0-387-00444-0 }}
{{cite book

| last = Bashirov
| first = Agamirza E
| title = Partially observable linear systems under dependent noises
| publisher = Basel; Boston: Birkhäuser Verlag
| year = 2003
| pages =
| isbn = 0-8176-6999-X
}}

{{cite book

| author2-link=Neil Trudinger
| first1=D.
| last1=Gilbarg
| first2=N.
| last2=Trudinger
| title = Elliptic partial differential equations of second order, 2nd ed
| publisher = Berlin; New York: Springer
| year = 2001
| pages =
| isbn = 3-540-41160-7
}}{{PlanetMath attribution|id=7154|title=Coercive Function}}

3 : Functional analysis|General topology|Types of functions

随便看

开放百科全书收录14589846条英语、德语、日语等多语种百科知识，基本涵盖了大多数领域的百科知识，是一部内容自由、开放的电子版国际百科全书。