User:Jjtorrens/BancoDePruebas
2020 Mathematics Subject Classification: Primary: 46E35 [MSN][ZBL]
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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 |
Jjtorrens/BancoDePruebas. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Jjtorrens/BancoDePruebas&oldid=26003