Generalisation of the derivative of a function
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
and
we have
![{\displaystyle {\begin{aligned}\int _{a}^{b}u(x)\varphi '(x)\,dx&={\Big [}u(x)\varphi (x){\Big ]}_{a}^{b}-\int _{a}^{b}u'(x)\varphi (x)\,dx.\\[6pt]\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9152c536ec17126a040beaddec24f7f51d61884b)
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 (
).
Definition
Let
be a function in the
Lebesgue space
. We say that
in
is a weak derivative of
if
![{\displaystyle \int _{a}^{b}u(t)\varphi '(t)\,dt=-\int _{a}^{b}v(t)\varphi (t)\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ec0eb67f2a094eeb247d1a21326b972efcd00ba)
for all infinitely
differentiable functions
with
.
Generalizing to
dimensions, if
and
are in the space
of
locally integrable functions for some
open set
, and if
is a
multi-index, we say that
is the
-weak derivative of
if
![{\displaystyle \int _{U}uD^{\alpha }\varphi =(-1)^{|\alpha |}\int _{U}v\varphi ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af12368a5394f5adb251b64f17315892f2dd0dc9)
for all
, that is, for all infinitely differentiable functions
with
compact support in
. Here
is defined as
![{\displaystyle D^{\alpha }\varphi ={\frac {\partial ^{|\alpha |}\varphi }{\partial x_{1}^{\alpha _{1}}\cdots \partial x_{n}^{\alpha _{n}}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f1dea5183714b95e69347cce1cf5141e5f739c4)
If
has a weak derivative, it is often written
since weak derivatives are unique (at least, up to a set of
measure zero, see below).
Examples
- The
absolute value function
, which is not differentiable at
has a weak derivative
known as the
sign function, and given by ![{\displaystyle v(t)={\begin{cases}1&{\text{if }}t>0;\\[6pt]0&{\text{if }}t=0;\\[6pt]-1&{\text{if }}t<0.\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3bc47bdd3cb90fe8efbe43a5450c4f31ef1b506f)
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. For example, the definition of v(0) above could be replaced with any desired real number. Usually, the existence of multiple solutions is not a problem, since functions are considered to be equivalent in the theory of
Lp spaces and
Sobolev spaces if they are equal almost everywhere.
- The
characteristic function of the rational numbers
is nowhere differentiable yet has a weak derivative. Since the
Lebesgue measure of the rational numbers is zero, ![{\displaystyle \int 1_{\mathbb {Q} }(t)\varphi (t)\,dt=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3537add576c3ae4935d5ab5be0618790eb786d3)
Thus
is a weak derivative of
. Note that this does agree with our intuition since when considered as a member of an Lp space,
is identified with the zero function.
- The
Cantor function c does not have a weak derivative, despite being differentiable almost everywhere. This is because any weak derivative of c would have to be equal almost everywhere to the classical derivative of c, which is zero almost everywhere. But the zero function is not a weak derivative of c, as can be seen by comparing against an appropriate test function
. More theoretically, c does not have a weak derivative because its
distributional derivative, namely the
Cantor distribution, is a
singular measure and therefore cannot be represented by a 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 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.
See also
References