I need to minimise $$\int\limits_\Omega|\nabla H_\epsilon(\phi)|\,dx\,dy$$ with respect to $\phi$. Where $H_\epsilon$ is the regularised Heaviside function, so that it is differentiable.
This can be rewritten as $$\int\limits_\Omega \delta_{\epsilon}(\phi)|\nabla \phi| \, dx \, dy.$$
The answer in this paper http://www.math.ucla.edu/~lvese/PAPERS/IJCV2002.pdf they say "Then we can formally write the associated Euler-Lagrange questions obtained by minimising the above functional as:"
$$-\delta_{\epsilon}\operatorname{div}\left(\frac{\nabla \phi}{|\nabla \phi|} \right). $$
Im not sure how this is obtained, i'm familiar with calculating the first variation but im unsure how to differentiate the second integrand.