Function of compact support

A function defined in some domain of $ E ^ {n} $, having compact support belonging to this domain. More precisely, suppose that the function $ f ( x) = f ( x _ {1} \dots x _ {n} ) $ is defined on a domain $ \Omega \subset E ^ {n} $. The support of $ f $ is the closure of the set of points $ x \in \Omega $ for which $ f ( x) $ is different from zero $ ( f ( x) \neq 0) $. Thus one can also say that a function of compact support in $ \Omega $ is a function defined on $ \Omega $ such that its support $ \Lambda $ is a closed bounded set located at a distance from the boundary $ \Gamma $ of $ \Omega $ by a number greater than $ \delta > 0 $, where $ \delta $ is sufficiently small.

One usually considers $ k $- times continuously-differentiable functions of compact support, where $ k $ is a given natural number. Even more often one considers infinitely-differentiable functions of compact support. The function

$$ \psi ( x) = \ \left \{ \begin{array}{ll} e ^ {- 1/( | x | ^ {2} - 1) } , & | x | < 1, | x | ^ {2} = \sum _ {j = 1 } ^ { n } x _ {j} ^ {2} , \\ 0 , & | x | \geq 1 , \\ \end{array} \right .$$

can serve as an example of an infinitely-differentiable function of compact support in a domain $ \Omega $ containing the sphere $ | x | \leq 1 $.

The set of all infinitely-differentiable functions of compact support in a domain $ \Omega \subset E ^ {n} $ is denoted by $ D $. On $ D $ one can define linear functionals (generalized functions, cf. Generalized function). With the aid of functions $ v \in D $ one can define generalized solutions (cf. Generalized solution) of boundary value problems.

In theorems concerned with problems on finding generalized solutions, it is often important to know whether $ D $ is dense in some concrete space of functions. It is known, for example, that if the boundary $ \Gamma $ of a bounded domain $ \Omega \subset E ^ {n} $ is sufficiently smooth, then $ D $ is dense in the space of functions

$$ {W ^ { o } } {} _ {p} ^ {r} ( \Omega ) = \ \left \{ {f } : { f \in W _ {p} ^ {r} ( \Omega ),\ \left . \frac{\partial ^ {s} f }{\partial n ^ {s} } \right | _ \Gamma = 0,\ s = 0 \dots r - 1 } \right \} $$

( $ 1 \leq p \leq \infty $), that is, in the Sobolev space of functions of class $ W _ {p} ^ {r} ( \Omega ) $ that vanish on $ \Gamma $ along with their normal derivatives of order up to and including $ r - 1 $( $ r = 1, 2 ,\dots $).

References

[1]	V.S. Vladimirov, "Generalized functions in mathematical physics" , MIR (1979) (Translated from Russian)
[2]	S.L. Sobolev, "Applications of functional analysis in mathematical physics" , Amer. Math. Soc. (1963) (Translated from Russian)

References

[a1]	L. Schwartz, "Théorie des distributions" , 1–2 , Hermann (1950–1951)
[a2]	J.L. Lions, E. Magenes, "Non-homogenous boundary value problems and applications" , 1–2 , Springer (1972) (Translated from French)
[a3]	K. Yosida, "Functional analysis" , Springer (1980) pp. Chapt. 8, Sect. 4; 5