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

$\langle Ax,x\rangle \geq c\|x\|^{2}$

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

$a(x,x)\geq c\|x\|^{2}$

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 ($|a(x,y)|\leq K\|x\|\,\|y\|$ 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. {{cite book}}: |edition= has extra text (help)CS1 maint: multiple names: authors list (link)

• 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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