Namespaces
Variants
Actions

User:Jjtorrens/BancoDePruebas

From Encyclopedia of Mathematics
Jump to: navigation, search
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.

2020 Mathematics Subject Classification: Primary: 46E35 [MSN][ZBL]


$\newcommand{\abs}[1]{\lvert #1\rvert} \newcommand{\norm}[1]{\lVert #1\rVert} \newcommand{\bfl}{\mathbf{l}}$ 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
How to Cite This Entry:
Jjtorrens/BancoDePruebas. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Jjtorrens/BancoDePruebas&oldid=26003