# User:Jjtorrens/BancoDePruebas

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

2010 Mathematics Subject Classification: Primary: 46E35 [MSN][ZBL]

$\newcommand{\abs}{\lvert #1\rvert} \newcommand{\norm}{\lVert #1\rVert} \newcommand{\bfl}{\mathbf{l}}$ 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 .

How to Cite This Entry:
Jjtorrens/BancoDePruebas. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Jjtorrens/BancoDePruebas&oldid=26003