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Weak derivative
From Wikipedia, the free encyclopedia
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 Lebesgue space L1([a,b]). See distributions for an even more general definition.
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Let u be a function in the Lebesgue space L1([a,b]). We say that v in L1([a,b]) is a weak derivative of u if,
for all continuously differentiable functions 
with 
.
Generalizing to n dimensions, if u and v are in the space 
of locally integrable functions for some open set 
, and if α is a multiindex, we say that v is the αth-weak derivative of u if
for all 
, that is, for all infinitely differentiable functions with compact support in U. If u has a weak derivative, it is often written Dαu since weak derivates are unique (at least, up to a set of measure zero, see below).
The function ![u:[-1,1]\to [0,1]:t\mapsto u(t)=|t|](../../../math/c/c/5/cc50ebc07a2f3a5fb1bf8e3dec0e45ab.png)
, which is not differentiable at t=0, has a weak derivative v known as the sign function given by :
![v :[-1,1]\to [-1,1]:t\mapsto v(t)=\left\{ { \begin{matrix} {1 \,\,\textrm{ if }\, t > 0} \\ {0 \,\, \textrm{ if }\, t=0} \\ {-1 \,\,\textrm{ if }\, t < 0} \\ \end{matrix} } \right.](../../../math/f/d/9/fd99495a0abb189e6305799a8851ee10.png)
It can be shown that 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, where 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 give 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 weak solutions in Sobolev spaces, which are useful for problems of differential equations and in functional analysis.