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

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

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. 

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