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Coercive vector fields

A vector field f : Rn → Rn is called coercive if f ( x ) ⋅ x ‖ x ‖ → + ∞  as  ‖ x ‖ → + ∞ , {\displaystyle {\frac {f(x)\cdot x}{\|x\|}}\to +\infty {\text{ as }}\|x\|\to +\infty ,} where " ⋅ {\displaystyle \cdot } " denotes the usual dot product and ‖ x ‖ {\displaystyle \|x\|} denotes the usual Euclidean norm of the vector x.

A coercive vector field is in particular norm-coercive since ‖ f ( x ) ‖ ≥ ( f ( x ) ⋅ x ) / ‖ x ‖ {\displaystyle \|f(x)\|\geq (f(x)\cdot x)/\|x\|} for x ∈ R n ∖ { 0 } {\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 f ( x ) ⋅ x = 0 {\displaystyle f(x)\cdot x=0} for every x ∈ R 2 {\displaystyle x\in \mathbb {R} ^{2}} .

Coercive operators and forms

A self-adjoint operator A : H → H , {\displaystyle A:H\to H,} where H {\displaystyle H} is a real Hilbert space, is called coercive if there exists a constant c > 0 {\displaystyle c>0} such that ⟨ A x , x ⟩ ≥ c ‖ x ‖ 2 {\displaystyle \langle Ax,x\rangle \geq c\|x\|^{2}} for all x {\displaystyle x} in H . {\displaystyle H.}

A bilinear form a : H × H → R {\displaystyle a:H\times H\to \mathbb {R} } is called coercive if there exists a constant c > 0 {\displaystyle c>0} such that a ( x , x ) ≥ c ‖ x ‖ 2 {\displaystyle a(x,x)\geq c\|x\|^{2}} for all x {\displaystyle x} in H . {\displaystyle H.}

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

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

If A : H → H {\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, ⟨ A x , x ⟩ ≥ C ‖ x ‖ {\displaystyle \langle Ax,x\rangle \geq C\|x\|} for big ‖ x ‖ {\displaystyle \|x\|} (if ‖ x ‖ {\displaystyle \|x\|} is bounded, then it readily follows); then replacing x {\displaystyle x} by x ‖ x ‖ − 2 {\displaystyle x\|x\|^{-2}} we get that A {\displaystyle A} is a coercive operator. One can also show that the converse holds true if A {\displaystyle A} is self-adjoint. The definitions of coercivity for vector fields, operators, and bilinear forms are closely related and compatible.

Norm-coercive mappings

A mapping f : X → X ′ {\displaystyle f:X\to X'} between two normed vector spaces ( X , ‖ ⋅ ‖ ) {\displaystyle (X,\|\cdot \|)} and ( X ′ , ‖ ⋅ ‖ ′ ) {\displaystyle (X',\|\cdot \|')} is called norm-coercive if and only if ‖ f ( x ) ‖ ′ → + ∞  as  ‖ x ‖ → + ∞ . {\displaystyle \|f(x)\|'\to +\infty {\mbox{ as }}\|x\|\to +\infty .}

More generally, a function f : X → X ′ {\displaystyle f:X\to X'} between two topological spaces X {\displaystyle X} and X ′ {\displaystyle X'} is called coercive if for every compact subset K ′ {\displaystyle K'} of X ′ {\displaystyle X'} there exists a compact subset K {\displaystyle K} of X {\displaystyle X} such that f ( X ∖ K ) ⊆ X ′ ∖ K ′ . {\displaystyle f(X\setminus K)\subseteq X'\setminus K'.}

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

(Extended valued) coercive functions

An (extended valued) function f : R n → R ∪ { − ∞ , + ∞ } {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} \cup \{-\infty ,+\infty \}} is called coercive if f ( x ) → + ∞  as  ‖ x ‖ → + ∞ . {\displaystyle f(x)\to +\infty {\mbox{ as }}\|x\|\to +\infty .} A real valued coercive function f : R n → R {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} } is, in particular, norm-coercive. However, a norm-coercive function f : R n → R {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} } is not necessarily coercive. For instance, the identity function on R {\displaystyle \mathbb {R} } is norm-coercive but not coercive.

See also

• 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.