Coercive function

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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[edit]

A vector field f : RnRn 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 : RnRn is not necessarily a coercive vector field. For instance the rotation f : R2R2, 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[edit]

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.

One can also show that any self-adjoint operator is a coercive operator if and only if 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). The definitions of coercivity for vector fields, operators, and bilinear forms are closely related and compatible.

Norm-coercive mappings[edit]

A mapping between two normed vectorspaces 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[edit]

An (extended valued) function is called coercive iff

A realvalued 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.

See also: radially unbounded functions

References[edit]

  • Renardy, Michael; Rogers, Robert C. (2004). An introduction to partial differential equations (Second ed.). New York, NY: Springer-Verlag. pp. xiv+434. ISBN 0-387-00444-0. 
  • Bashirov, Agamirza E (2003). Partially observable linear systems under dependent noises. Basel; Boston: Birkhäuser Verlag. ISBN 0-8176-6999-X. 
  • Gilbarg, D.; Trudinger, N. (2001). Elliptic partial differential equations of second order, 2nd ed. Berlin; New York: Springer. ISBN 3-540-41160-7. 

This article incorporates material from Coercive Function on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.