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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 : RnRn is called coercive if

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

where "$cdot$" denotes the usual dot product and $|x|$ denotes the usual Euclidean norm of the vector x.

More generally, a function f : XY 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

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

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

Coercive operators and forms

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

$langle Ax, xrangle ge c|x|^2$

for all $x$ in $H.$

A bilinear form $a:Htimes Hto mathbb R$ is called coercive if there exists a constant $c>0$ such that

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

for all $x$ in $H.$

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

$a\left(x, y\right)=langle Ax, yrangle$

for some self-adjoint operator $A:Hto H,$ 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 $A:Hto H$ 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, 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.

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