How can we prove that for any $\phi_n\to \phi$ in $W^{1,1}(\Omega)$ then there exist always:
$$\lim\limits_{n\to\infty} \int_{\Omega}|\nabla\phi_n(x)| dx = \int_{\Omega} |\nabla\phi(x)|\ dx$$
?
The $\geq$ inequality is well-known from Fatou Lemma. I tried some examples and the equality seem to hold. Maybe the following links will help: The total variation of functions in the Sobolev space $W^{1,1}$ and Gradient inequality lower semicontinuity
Motivation
I didn't find in any textbook this kind of properties. It will be very useful for perimeter problems since the integral of the gradient norm is the variation of the function which is closely related to perimeter.
It is more simple. Since $\phi_n \to \phi$ in $W^{1,1}$, we have in particular:
$$\int_\Omega |\nabla \phi_n - \nabla \phi| \to 0$$
Since $ ||\nabla \phi_n| - |\nabla \phi|| \le |\nabla \phi_n - \nabla \phi|$, we deduce
$$\int_\Omega \left( |\nabla \phi_n| - |\nabla \phi| \right)\to 0$$
Finally, we obtain
$$\int_\Omega |\nabla \phi_n| \to \int_\Omega |\nabla \phi|$$