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
The method of integration by parts holds that for differentiable functions
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
vanishing at the boundary points (
be a function in the Lebesgue space
is a weak derivative of
if for all infinitely differentiable functions
dimensions, if
are in the space
loc
{\displaystyle L_{\text{loc}}^{1}(U)}
of locally integrable functions for some open set
is a multi-index, we say that
-weak derivative of
, that is, for all infinitely differentiable functions
with compact support in
is defined as
has a weak derivative, it is often written
since weak derivatives are unique (at least, up to a set of measure zero, see below) [3].
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.
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.