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Concept
Weak derivative
Summary
In mathematics, a weak derivative is a generalization of the concept of the derivative of a function (strong derivative) for functions not assumed differentiable, but only integrable, i.e., to lie in the Lp space .
The method of integration by parts holds that for differentiable functions and we have
A function u' being the weak derivative of u is essentially defined by the requirement that this equation must hold for all infinitely differentiable functions φ vanishing at the boundary points ().
Let be a function in the Lebesgue space . We say that in is a weak derivative of if
for all infinitely differentiable functions with .
Generalizing to dimensions, if and are in the space of locally integrable functions for some open set , and if is a multi-index, we say that is the -weak derivative of if
for all , that is, for all infinitely differentiable functions with compact support in . Here is defined as
If has a weak derivative, it is often written since weak derivatives are unique (at least, up to a set of measure zero, see below).
The absolute value function , which is not differentiable at has a weak derivative known as the sign function, and given by This is not the only weak derivative for u: any w that is equal to v almost everywhere is also a weak derivative for u. (In particular, the definition of v(0) above is superfluous and can be replaced with any desired real number r.) Usually, this is not a problem, since in the theory of Lp spaces and Sobolev spaces, functions that are equal almost everywhere are identified.
The characteristic function of the rational numbers is nowhere differentiable yet has a weak derivative. Since the Lebesgue measure of the rational numbers is zero, Thus is a weak derivative of . Note that this does agree with our intuition since when considered as a member of an Lp space, is identified with the zero function.
The Cantor function c does not have a weak derivative, despite being differentiable almost everywhere.
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