# weak derivative

Let $f\colon\Omega\to\mathbf{R}$ and $g=(g_{1},\ldots,g_{n})\colon\Omega\to\mathbf{R}^{n}$ be locally integrable functions defined on an open set $\Omega\subset\mathbf{R}^{n}$. We say that $g$ is the weak derivative of $f$ if the equality

 $\int_{\Omega}f\frac{\partial\phi}{\partial x_{i}}=-\int_{\Omega}g_{i}\phi$

holds true for all functions $\phi\in\mathcal{C}^{\infty}_{c}(\Omega)$ (smooth functions with compact support in $\Omega$) and for all $i=1,\ldots,n$. Notice that the integrals involved are well defined since $\phi$ is bounded with compact support and because $f$ is assumed to be integrable on compact subsets of $\Omega$.

1. 1.

If $f$ is of class $\mathcal{C}^{1}$ then the gradient of $f$ is the weak derivative of $f$ in view of Gauss Green Theorem. So the weak derivative is an extension of the classical derivative.

2. 2.

The weak derivative is unique (as an element of the Lebesgue space $L^{1}_{\mathrm{l}oc}$) in view of a result about locally integrable functions.

3. 3.

The same definition can be given for functions with complex values.

Title weak derivative WeakDerivative 2013-03-22 14:54:52 2013-03-22 14:54:52 paolini (1187) paolini (1187) 16 paolini (1187) Definition msc 46E35 SobolevSpaces