Editing Weak derivative

{{Short description|Generalisation of the derivative of a function}}
{{more footnotes|date=May 2014}}

In [[mathematics]], a '''weak derivative''' is a generalization of the concept of the [[derivative]] of a [[function (mathematics)|function]] (''strong derivative'') for functions not assumed [[Differentiable function|differentiable]], but only [[Integrable function|integrable]], i.e., to lie in the [[Lp space|L<sup>''p''</sup> space]] <math>L^1([a,b])</math>.

The method of [[integration by parts]] holds that for smooth functions <math>u</math> and <math>\varphi</math> we have

:<math display=block>\begin{align}
   \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{align}</math>

A function ''u''<nowiki/>' being the weak derivative of ''u'' is essentially defined by the requirement that this equation must hold for all smooth functions <math>\varphi</math> vanishing at the boundary points (<math>\varphi(a)=\varphi(b)=0</math>).

== Definition ==

Let <math>u</math> be a function in the [[Lp space|Lebesgue space]] <math>L^1([a,b])</math>. We say that <math>v</math>   in <math>L^1([a,b])</math> is a '''weak derivative''' of <math>u</math> if

:<math>\int_a^b u(t)\varphi'(t) \, dt=-\int_a^b v(t)\varphi(t) \, dt</math>

for ''all'' infinitely [[differentiable function]]s <math> \varphi </math> with <math>\varphi(a)=\varphi(b)=0</math>.<ref>{{Cite book |last=Evans |first=Lawrence C. |title=Partial differential equations |date=1998 |publisher=American mathematical society |isbn=978-0-8218-0772-9 |series=Graduate studies in mathematics |location=Providence (R. I.) |pages=242}}</ref><ref>{{Cite book |last=Gilbarg |first=David |title=Elliptic partial differential equations of second order |last2=Trudinger |first2=Neil S. |date=2001 |publisher=Springer |isbn=978-3-540-41160-4 |edition=2nd ed., rev. 3rd printing |series=Classics in mathematics |location=Berlin New York |pages=149}}</ref>

Generalizing to <math>n</math> dimensions, if <math>u</math> and <math>v</math> are in the space <math>L_\text{loc}^1(U)</math> of [[locally integrable function]]s for some [[open set]] <math>U \subset \mathbb{R}^n</math>, and if <math>\alpha</math> is a [[multi-index]], we say that <math>v</math> is the <math>\alpha^\text{th}</math>-weak derivative of <math>u</math> if

:<math>\int_U u D^\alpha \varphi=(-1)^{|\alpha|} \int_U v\varphi,</math>

for all <math>\varphi \in C^\infty_c (U)</math>, that is, for all infinitely differentiable functions <math>\varphi</math> with [[compact support]] in <math>U</math>. Here <math> D^{\alpha}\varphi</math> is defined as
<math display="block"> D^{\alpha}\varphi = \frac{\partial^{| \alpha |} \varphi }{\partial x_1^{\alpha_1} \cdots \partial x_n^{\alpha_n}}.</math>

If <math>u</math> has a weak derivative, it is often written <math>D^{\alpha}u</math> since weak derivatives are unique (at least, up to a set of [[measure zero]], see below).<ref>{{Cite book |last=Knabner |first=Peter |title=Numerical Methods for Elliptic and Parabolic Partial Differential Equations |last2=Angermann |first2=Lutz |date=2003 |publisher=Springer New York |isbn=978-0-387-95449-3 |series=Texts in Applied Mathematics |location=New York, NY |pages=53}}</ref>

== Examples ==
*The [[absolute value]] function <math>u : \mathbb{R} \rightarrow \mathbb{R}_+, u(t) = |t|</math>, which is not differentiable at <math>t = 0</math> has a weak derivative <math>v: \mathbb{R} \rightarrow \mathbb{R}</math> known as the [[sign function]], and given by <math display="block">
v(t) = \begin{cases} 1 & \text{if } t > 0; \\[6pt] 0 & \text{if } t = 0; \\[6pt] -1 & \text{if } t < 0. \end{cases}</math> 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 space|''L''<sup>''p''</sup> spaces]] and [[Sobolev space]]s if they are equal almost everywhere.
*The [[indicator function|characteristic function]] of the rational numbers <math> 1_{\mathbb{Q}} </math> is nowhere differentiable yet has a weak derivative.  Since the [[Lebesgue measure]] of the rational numbers is zero, <math display="block"> \int 1_{\mathbb{Q}}(t) \varphi(t) \, dt = 0.</math> Thus <math> v(t)=0 </math> is a weak derivative of <math> 1_{\mathbb{Q}} </math>.  Note that this does agree with our intuition since when considered as a member of an Lp space, <math> 1_{\mathbb{Q}} </math> 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 <math>\varphi</math>. 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 solution]]s in [[Sobolev space]]s, which are useful for problems of [[differential equations]] and in [[functional analysis]].

==See also==
*[[Subderivative]]
*[[Weyl's lemma (Laplace equation)]]

==References==
{{Reflist}}

[[Category:Generalized functions]]
[[Category:Functional analysis]]
[[Category:Generalizations of the derivative]]
[[Category:Generalizations]]