weak derivative
Let and be locally integrable functions defined on an open set . We say that is the weak derivative of if the equality
holds true for all functions (smooth functions with compact support in ) and for all . Notice that the integrals involved are well defined since is bounded with compact support and because is assumed to be integrable on compact subsets of .
Comments
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1.
If is of class then the gradient of is the weak derivative of in view of Gauss Green Theorem. So the weak derivative is an extension of the classical derivative.
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2.
The weak derivative is unique (as an element of the Lebesgue space ) in view of a result about locally integrable functions.
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3.
The same definition can be given for functions with complex values.
Title | weak derivative |
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Canonical name | WeakDerivative |
Date of creation | 2013-03-22 14:54:52 |
Last modified on | 2013-03-22 14:54:52 |
Owner | paolini (1187) |
Last modified by | paolini (1187) |
Numerical id | 16 |
Author | paolini (1187) |
Entry type | Definition |
Classification | msc 46E35 |
Related topic | SobolevSpaces |