Weak derivative

From Wikipedia, the free encyclopedia

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 $L 1 ([a, b])$ . See distributions for an even more general definition.

1 Definition
2 Examples
3 Properties
4 Extensions

[edit] Definition

Let $u$ be a function in the Lebesgue space $L 1 ([a, b])$ . We say that $v$ in $L 1 ([a, b])$ is a weak derivative of $u$ if,

$\int_a^b u(t)\varphi'(t)dt=-\int_a^b v(t)\varphi(t)dt$

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

Generalizing to $n$ dimensions, if $u$ and $v$ are in the space $L_{loc}^1(U)$ of locally integrable functions for some open set $U \subset \mathbb{R}^n$ , and if $α$ is a multiindex, we say that $v$ is the $α t h$ -weak derivative of $u$ if

$\int_U u D^{\alpha} \varphi=(-1)^{|\alpha|} \int_U v\varphi$

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

[edit] Examples

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

$v :[-1,1]\to [-1,1]:t\mapsto v(t)=\left\{ { \begin{matrix} {1 \,\,\textrm{ if }\, t > 0} \\ {0 \,\, \textrm{ if }\, t=0} \\ {-1 \,\,\textrm{ if }\, t < 0} \\ \end{matrix} } \right.$

[edit] 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.