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 L^1([a,b]). The method of integration by parts holds that for differentiable functions u and \varphi 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 \varphi vanishing at the boundary points.

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 for all infinitely differentiable functions \varphi with. Generalizing to n dimensions, if u and v are in the space of locally integrable functions for some open set, and if \alpha is a multi-index, we say that v is the -weak derivative of u if for all, that is, for all infinitely differentiable functions \varphi with compact support in U. Here is defined as If u has a weak derivative, it is often written D^{\alpha}u since weak derivatives are unique (at least, up to a set of measure zero, see below).

Examples

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 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 (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.