I am considering the following minimizing problem:
$$
\min\_{u\in BV(\Omega)}\{\frac12\|u-u\_0\|\_{L^2}^2 + |u|\_{TV(\Omega)}\}
$$
where $u\_0\in BV(\Omega)$, $\Omega\subset \mathbb R^2$ is open bounded, smooth boundary.


The above problem surely has a unique minimizer $\bar u\in BV$.


Then I could write the corresponding Euler-Lagrange equation for $\bar u$:
$$
\bar u - \operatorname{div}\left(\frac{\nabla \bar u}{|\nabla \bar u|}\right) = u\_0 \tag 1
$$
Next, I define the operator $A$ to be
$$
A(u):=- \operatorname{div}\left(\frac{\nabla u}{|\nabla u|}\right) 
$$
I am wishing to show $A$ is a monotone operator. It is easy to see that for $v\in C\_0^\infty$, that $(Av,v)\geq 0$ for sure. 


My question: what should be the right domain for operator $A$? I think the domain of $A$ should somehow contain the space $BV$ since it is the E-L for minimizing function, but a $BV$ function wouldn't make sense for equation $(1)$. Maybe there are some regularity result I am missing?



The monotone operator in $BV$ space
$\varphi:=|.|\_{TV}$ is a convex and (I think) lower semicontinuous function from $L^2$ to $[0,\infty]$. Its subdifferential $\partial\varphi$ is what you need to write the Euler-Lagrange equation, in the form $\bar{u}-u\_0\in\partial\varphi(\bar{u})$.


