, a coercive function
is a function that "grows rapidly" at the extremes of the space on which it is defined. More precisely, a function f
is called coercive
where "" denotes the usual dot product and denotes the usual Euclidean norm of the vector x.
More generally, a function f : X → Y between two topological spaces X and Y is called coercive if for every compact subset J of Y there exists a compact subset K of X such that
The composition of a bijective proper map followed by a coercive map is coercive.
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 ( 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 operator 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 function (if one replaces the dot product with the more general inner product in the definition of coercivity of a function). The definitions of coercivity for functions, operators, and bilinear forms are closely related and compatible.
- Renardy, Michael and Rogers, Robert C. (2004). An introduction to partial differential equations. Second edition, New York, NY: Springer-Verlag. ISBN 0-387-00444-0.
- Bashirov, Agamirza E Partially observable linear systems under dependent noises. Basel; Boston: Birkhäuser Verlag.
- Gilbarg, David; Trudinger, Neil S. Elliptic partial differential equations of second order, 2nd ed. Berlin; New York: Springer.