Coercive function
Mathematical function From Wikipedia, the free encyclopedia
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.
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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
Summarize
Perspective
A mapping between two normed vector spaces and is called norm-coercive if and only if
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 if 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.
See also
References
- 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.
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