Sobolev generalized derivative

A locally summable generalized derivative of a locally summable function (see Generalized function).

More explicitly, if is an open set in an -dimensional space and if and are locally summable functions on , then is the Sobolev generalized partial derivative with respect to of on :

if the following equation holds:

for all infinitely-differentiable functions on with compact support. The Sobolev generalized derivative is only defined almost-everywhere on .

An equivalent definition is as follows: Suppose that a locally summable function on can be modified in such a way that, on a set of -dimensional measure zero, it will be locally absolutely continuous with respect to for almost-all points , in the sense of the -dimensional measure. Then has an ordinary partial derivative with respect to for almost-all . If the latter is locally summable, then it is called a Sobolev generalized derivative.

A third equivalent definition is as follows: Given two functions and , suppose there is a sequence of continuously-differentiable functions on such that for any domain whose closure lies in ,

Then is the Sobolev generalized derivative of on .

Sobolev generalized derivatives of on of higher orders (if they exist) are defined inductively:

They do not depend on the order of differentiation; e.g.,

almost-everywhere on .

References

[1]	S.L. Sobolev, "Some applications of functional analysis in mathematical physics" , Amer. Math. Soc. (1963) (Translated from Russian)
[2]	S.M. Nikol'skii, "A course of mathematical analysis" , 2 , MIR (1977) (Translated from Russian)

Comments

In the Western literature the Sobolev generalized derivative is called the weak or distributional derivative.

References

[a1]	L. Schwartz, "Théorie des distributions" , 1–2 , Hermann (1973)
[a2]	K. Yosida, "Functional analysis" , Springer (1980) pp. Chapt. 8, Sect. 4; 5