Coercive function

A vector field f : Rⁿ → Rⁿ is called coercive if

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

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

A coercive vector field is in particular norm-coercive since ${\displaystyle \|f(x)\|\geq (f(x)\cdot x)/\|x\|}$ for ${\displaystyle x\in \mathbb {R} ^{n}\setminus \{0\}}$, by Cauchy–Schwarz inequality. However a norm-coercive mapping f : Rⁿ → Rⁿ is not necessarily a coercive vector field. For instance the rotation f : R² → R², f(x) = (−x₂, x₁) by 90° is a norm-coercive mapping which fails to be a coercive vector field since ${\displaystyle f(x)\cdot x=0}$ for every ${\displaystyle x\in \mathbb {R} ^{2}}$.

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

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

for all ${\displaystyle x}$ in ${\displaystyle H.}$

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

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

for all ${\displaystyle x}$ in ${\displaystyle H.}$

It follows from the Riesz representation theorem that any symmetric (defined as ${\displaystyle a(x,y)=a(y,x)}$ for all ${\displaystyle x,y}$ in ${\displaystyle H}$), continuous (${\displaystyle |a(x,y)|\leq k\|x\|\,\|y\|}$ for all ${\displaystyle x,y}$ in ${\displaystyle H}$ and some constant ${\displaystyle k>0}$) and coercive bilinear form ${\displaystyle a}$ has the representation

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

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

If ${\displaystyle A:H\to H}$ is a coercive operator then it is a coercive mapping (in the sense of coercivity of a vector field, where one has to replace the dot product with the more general inner product). Indeed, ${\displaystyle \langle Ax,x\rangle \geq C\|x\|}$ for big ${\displaystyle \|x\|}$ (if ${\displaystyle \|x\|}$ is bounded, then it readily follows); then replacing ${\displaystyle x}$ by ${\displaystyle x\|x\|^{-2}}$ we get that ${\displaystyle A}$ is a coercive operator. One can also show that the converse holds true if ${\displaystyle A}$ is self-adjoint. The definitions of coercivity for vector fields, operators, and bilinear forms are closely related and compatible.

A mapping ${\displaystyle f:X\to X'}$ between two normed vector spaces ${\displaystyle (X,\|\cdot \|)}$ and ${\displaystyle (X',\|\cdot \|')}$ is called norm-coercive if and only if

$${\displaystyle \|f(x)\|'\to +\infty {\mbox{ as }}\|x\|\to +\infty .}$$

More generally, a function ${\displaystyle f:X\to X'}$ between two topological spaces ${\displaystyle X}$ and ${\displaystyle X'}$ is called coercive if for every compact subset ${\displaystyle K'}$ of ${\displaystyle X'}$ there exists a compact subset ${\displaystyle K}$ of ${\displaystyle X}$ such that

$${\displaystyle f(X\setminus K)\subseteq X'\setminus K'.}$$

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

An (extended valued) function ${\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} \cup \{-\infty ,+\infty \}}$ is called coercive if

$${\displaystyle f(x)\to +\infty {\mbox{ as }}\|x\|\to +\infty .}$$

A real valued coercive function ${\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }$ is, in particular, norm-coercive. However, a norm-coercive function ${\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }$ is not necessarily coercive. For instance, the identity function on ${\displaystyle \mathbb {R} }$ is norm-coercive but not coercive.

• Radially unbounded functions

• Renardy, Michael; Rogers, Robert C. (2004). An introduction to partial differential equations (Second 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 0-8176-6999-X.
• Gilbarg, D.; Trudinger, N. (2001). Elliptic partial differential equations of second order, 2nd ed. Berlin; New York: Springer. ISBN 3-540-41160-7.

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