Coercive function

In mathematics, 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 : Rⁿ → Rⁿ is called coercive if

$\frac{f(x) \cdot x}{\| x \|} \to %2B \infty \mbox{ as } \| x \| \to %2B \infty,$

where " $\cdot$ " denotes the usual dot product and $\|x\|$ 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

$f (X \setminus K) \subseteq Y \setminus J.$

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

Coercive operators and forms

A self-adjoint operator $A:H\to H,$ where $H$ is a real Hilbert space, is called coercive if there exists a constant $c>0$ such that

$\langle Ax, x\rangle \ge c\|x\|^2$

for all $x$ in $H.$

A bilinear form $a:H\times H\to \mathbb R$ is called coercive if there exists a constant $c>0$ such that

$a(x, x)\ge c\|x\|^2$

for all $x$ in $H.$

It follows from the Riesz representation theorem that any symmetric ( $a(x, y)=a(y, x)$ for all $x, y$ in $H$ ), continuous ( $|a(x, y)|\le K\|x\|\,\|y\|$ for all $x, y$ in $H$ and some constant $K>0$ ) and coercive bilinear form $a$ has the representation

$a(x, y)=\langle Ax, y\rangle$

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

One can also show that any self-adjoint operator $A:H\to H$ 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.

References

Renardy, Michael and Rogers, Robert C. (2004). An introduction to partial differential equations (Second edition 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 081766999X.

Gilbarg, D.; Trudinger, N. (2001). Elliptic partial differential equations of second order, 2nd ed. Berlin; New York: Springer. ISBN 3540411607.

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