weak derivative

Let $f:\mathrm{\Omega }\to 𝐑$ and $g=(g_1,\mathrm{\dots },g_n):\mathrm{\Omega }\to {𝐑}^n$ be locally integrablefunctions defined on an open set $\mathrm{\Omega }\subset {𝐑}^n$.We say that $g$ is the weak derivative of $f$ if the equality

$${\int }_{\mathrm{\Omega }}f\frac{\partial \varphi }{\partial x_i}=-{\int }_{\mathrm{\Omega }}{g}_i\varphi$$

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

Comments

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

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