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 Lebesgue space $L^1(*)$. See distributions for an even more general definition.

Definition

Let $u$ be a function in the Lebesgue space is a weak derivative of $u$ if,

$\backslash int\_a^b\; u(t)\backslash varphi\text{'}(t)dt=-\backslash int\_a^b\; v(t)\backslash varphi(t)dt$

for all continuously differentiable functions $\backslash varphi$ with $\backslash varphi(a)=\backslash varphi(b)=0$.

Generalizing to $n$ dimensions, if $u$ and $v$ are in the space $L\_\{loc\}^1(U)$ of locally integrable functions for some open set $U\; \backslash subset\; \backslash mathbb\{R\}^n$, and if $\backslash alpha$ is a multiindex, we say that $v$ is the $\backslash alpha^\{th\}$-weak derrivative of $u$ if

$\backslash int\_U\; u\; D^\{\backslash alpha\}\; \backslash varphi=(-1)^\{|\backslash alpha|\}\; \backslash int\_U\; v\backslash varphi$

for all $\backslash varphi\; \backslash in\; C^\{\backslash infty\}\_c\; (U)$, that is, for all infinitely differentiable functions $\backslash varphi$ with compact support in $U$. If $u$ has a weak derivative, it is often written $D^\{\backslash alpha\}u$ since weak derivates are unique (at least, up to a set of measure zero, see below).

Examples

The function $u:[0,1:t\backslash mapsto\; u(t)=|t|$, which is not differentiable at t=0, has a weak derivative v known as the sign function given by :

Properties

It can be shown that 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 give 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 weak solutions in Sobolev spaces, which are useful for problems of differential equations and in functional analysis.

Functional analysis | Schwache Ableitung