---
tag: W8W9AP79V5
kind: definition
name: Weak derivative
msc: ["46E35", 35D30]
defines: C3EC148VZD
uses: [09EGC6F8PT, DWJF7ABPR1]
sources:
  - url: https://en.wikipedia.org/wiki/Weak_derivative
---

## Definition

Let $U \subseteq \R^n$ be open.
A function $u \colon U \to \R$ is *locally integrable* if it is measurable and $\int_K \abs{u} \, dx < \infty$ for every compact set $K \subset U$.
Let $C_c^\infty(U)$ be the space of smooth functions on $U$ with compact support in $U$, the test functions.
For a multi-index $\alpha = (\alpha_1, \ldots, \alpha_n)$ write $\abs{\alpha} = \alpha_1 + \cdots + \alpha_n$ and $\partial^\alpha = \partial_1^{\alpha_1} \cdots \partial_n^{\alpha_n}$.
Let $u$ and $v$ be locally integrable on $U$.
The function $v$ is the *$\alpha$-th weak derivative* of $u$, written $D^\alpha u = v$, if
$$\int_U u \, \partial^\alpha \varphi \, dx = (-1)^{\abs{\alpha}} \int_U v \, \varphi \, dx \quad \text{for all } \varphi \in C_c^\infty(U).$$

## Remarks

The weak derivative, when it exists, is unique up to a set of measure zero.
If $u$ is of class $C^{\abs{\alpha}}$ on $U$, then its classical [[partial derivative|09EGC6F8PT]] $\partial^\alpha u$ is also its weak derivative, by integration by parts.

Example: on $U = (-1, 1)$, the function $u(x) = \abs{x}$ has weak derivative $v = \sgn x$.
For $\varphi \in C_c^\infty(-1, 1)$, integration by parts on $(-1, 0)$ and on $(0, 1)$ gives $\int_{-1}^{1} \abs{x} \varphi'(x) \, dx = \int_{-1}^{0} \varphi \, dx - \int_{0}^{1} \varphi \, dx = -\int_{-1}^{1} \sgn(x) \varphi(x) \, dx$, since the boundary terms at $0$ vanish.

Non-example: on $(-1, 1)$, the function $\sgn x$ has no weak derivative.
For any $\varphi \in C_c^\infty(-1, 1)$ one finds $\int \sgn(x) \varphi'(x) \, dx = -2 \varphi(0)$.
A locally integrable $v$ with $\int v \varphi \, dx = 2 \varphi(0)$ for all $\varphi$ would vanish almost everywhere on $(-1, 0)$ and on $(0, 1)$, and then the integral would be $0$, not $2 \varphi(0)$.
In the language of distributions, the derivative of $\sgn x$ is $2\delta$, which is not a function.