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?