Weak derivative

 ( Generalized Functions ) 

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 $L^1(a,b)$.

The method of integration by parts holds that for smooth functions $u$ and $\backslash varphi$ we have

$$\backslash begin\{align\}$$

  \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}
     

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

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Definition Let $u$ be a function in the Lp space $L^1(a,b)$. We say that $v$ in $L^1(a,b)$ is a weak derivative of $u$ if

$\backslash int\_a^b\; u(t)\backslash varphi\text{'}(t)\; \backslash ,\; dt=-\backslash int\_a^b\; v(t)\backslash varphi(t)\; \backslash ,\; dt$

for all infinitely differentiable functions $\backslash varphi$ with $\backslash varphi(a)=\backslash varphi(b)=0$.

David Gilbarg (2025). 9783540411604, Springer. ISBN 9783540411604

Generalizing to $n$ dimensions, if $u$ and $v$ are in the space $L\_\backslash text\{loc\}^1(U)$ of locally integrable functions for some open set $U\; \backslash subset\; \backslash mathbb\{R\}^n$, and if $\backslash alpha$ is a multi-index, we say that $v$ is the $\backslash alpha^\backslash text\{th\}$-weak derivative of $u$ if

$\backslash int\_U\; u\; D^\backslash alpha\; \backslash varphi=(-1)^$ \int_U v\varphi,

for all $\backslash varphi\; \backslash in\; C^\backslash infty\_c\; (U)$, that is, for all infinitely differentiable functions $\backslash varphi$ with compact support in $U$. Here $D^\{\backslash alpha\}\backslash varphi$ is defined as $$D^\{\backslash alpha\}\backslash varphi\; =\; \backslash frac\{\backslash partial^$$

\varphi }{\partial x_1^{\alpha_1} \cdots \partial x_n^{\alpha_n}}.

If $u$ has a weak derivative, it is often written $D^\{\backslash alpha\}u$ since weak derivatives are unique (at least, up to a set of measure zero, see below).

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Examples

• The absolute value function $u\; :\; \backslash mathbb\{R\}\; \backslash rightarrow\; \backslash mathbb\{R\}\_+,\; u(t)\; =\; |t|$, which is not differentiable at $t\; =\; 0$ has a weak derivative $v:\; \backslash mathbb\{R\}\; \backslash rightarrow\; \backslash mathbb\{R\}$ known as the sign function, and given by $$$$

v(t) = \begin{cases} 1 & \text{if } t > 0; \\6pt 0 & \text{if } t = 0; \\6pt -1 & \text{if } t < 0. \end{cases} 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 and if they are equal almost everywhere.

• The characteristic function of the rational numbers $1\_\{\backslash mathbb\{Q\}\}$ is nowhere differentiable yet has a weak derivative. Since the Lebesgue measure of the rational numbers is zero, $$\backslash int\; 1\_\{\backslash mathbb\{Q\}\}(t)\; \backslash varphi(t)\; \backslash ,\; dt\; =\; 0.$$ Thus $v(t)=0$ is a weak derivative of $1\_\{\backslash mathbb\{Q\}\}$. Note that this does agree with our intuition since when considered as a member of an Lp space, $1\_\{\backslash mathbb\{Q\}\}$ 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 $\backslash varphi$. 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.

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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.

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Extensions This concept gives rise to the definition of in , which are useful for problems of differential equations and in functional analysis.

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See also

• Subderivative
• Weyl's lemma (Laplace equation)

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