Editing Weak derivative (section)

== 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>