Weak derivative - Imported Knowlpedia Content

Definition

Let $\Omega\subseteq\mathbb R^n$ be open and let $f$ be locally integrable, meaning Lebesgue integrable on every compact subset of $\Omega$ . A locally integrable function $g$ is the weak partial derivative $\partial_j f$ if

\int_\Omega f\,\partial_j\varphi\,dx =-\int_\Omega g\,\varphi\,dx

for every $\varphi\in C_c^\infty(\Omega)$ . Equivalently, the distributional derivative of the regular distribution defined by $f$ is the regular distribution defined by $g$ . A weak derivative, when it exists, is unique up to equality almost everywhere.

Agreement with classical derivatives

If $f\in C^1(\Omega)$ , integration by parts shows that its classical partial derivative is also its weak derivative. More generally, a locally absolutely continuous function on an interval has a weak derivative equal almost everywhere to its ordinary derivative. Weak differentiation therefore extends rather than replaces classical differentiation.

Existence and failure

The absolute-value function on $\mathbb R$ has weak derivative $\operatorname{sgn}(x)$ , even though it is not differentiable at $0$ . The Heaviside function has no weak derivative represented by a locally integrable function: its distributional derivative is $\delta_0$ . Thus every locally integrable function has a distributional derivative, but not every such derivative is weak in the function-valued sense.

Sobolev-space role

A function belongs to the Sobolev space $W^{1,p}(\Omega)$ when it and all its first weak derivatives belong to $L^p(\Omega)$ ; higher-order Sobolev spaces use weak derivatives indexed by multi-indices. This makes differentiability compatible with norm completion and variational methods Evans, §5.2.

References

Lawrence C. Evans, Partial Differential Equations, 2nd ed., American Mathematical Society, 2010. DOI record. Relevant: §5.2 on weak derivatives and Sobolev spaces.
Robert A. Adams and John J. F. Fournier, Sobolev Spaces, 2nd ed., Academic Press, 2003. Publisher record. Relevant: Chapter 3 on weak derivatives.