Given two sufficiently regular functions $u,v$ on a domain $\Omega$ and a functional $F[u] := \int_\Omega \min(u,0)dx$.
Now I want to determine the functional derivative of $F$. Therefore
$DF[u]v = \frac{d}{d\epsilon}\big|_{\epsilon = 0} \int_\Omega \min(u+\epsilon v,0)dx$.
My result would be $DF[u|v = \int_\Omega \mathbb{1}_{u<0}vdx$.
Is this the correct solution ? I'm a bit insecure about it. Thank you very much.