# Weak derivative

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 $\mathrm {L} ^{1}([a,b])$ . See distributions for an even more general definition.

## Definition

$\int _{a}^{b}u(t)\varphi '(t)dt=-\int _{a}^{b}v(t)\varphi (t)dt$ $\int _{U}uD^{\alpha }\varphi =(-1)^{|\alpha |}\int _{U}v\varphi$ ## Examples

• The absolute value function u : [−1, 1] → [0, 1], u(t) = |t|, which is not differentiable at t = 0, has a weak derivative v known as the sign function given by
$v\colon [-1,1]\to [-1,1]\colon t\mapsto v(t)={\begin{cases}1,&{\mbox{if }}t>0;\\0,&{\mbox{if }}t=0;\\-1,&{\mbox{if }}t<0.\end{cases}}$ 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. Usually, this is not a problem, since in the theory of Lp spaces and Sobolev spaces, functions that are equal almost everywhere are identified.
$\int 1_{\mathbb {Q} }(t)\varphi (t)dt=0.$ Thus $v(t)=0$ is the weak derivative of $1_{\mathbb {Q} }$ . Note that this does agree with our intuition since when considered as a member of an Lp space, $1_{\mathbb {Q} }$ is identified with the zero function.

## Properties

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, where 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 (in the sense given above) to its conventional (strong) 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.