Weak derivative


In mathematics, a weak derivative is a generalization of the concept of the derivative of a function for functions not assumed differentiable, but only integrable, i.e., to lie in the Lp space. See distributions for a more general definition.

Definition

Let be a function in the Lebesgue space. We say that in is a weak derivative of if
for all infinitely differentiable functions with. This definition is motivated by the integration technique of integration by parts.
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.

Examples

If two functions are weak derivatives of the same function, they are equal except on a set with Lebesgue measure zero, i.e., they are equal almost everywhere. If we consider equivalence classes of functions such that two functions are equivalent if they are equal almost everywhere, then the weak derivative is unique.
Also, if u is differentiable in the conventional sense then its weak derivative is identical to its conventional derivative. Thus the weak derivative is a generalization of the strong one. Furthermore, the classical rules for derivatives of sums and products of functions also hold for the weak derivative.

Extensions

This concept gives rise to the definition of weak solutions in Sobolev spaces, which are useful for problems of differential equations and in functional analysis.