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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?

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$\varphi:=|.|_{TV}$ 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 $\bar{u}-u_0\in\partial\varphi(\bar{u})$.

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  • $\begingroup$ So, do you think, in your suggestion, that $A$ is an maximal monotone operator? $\endgroup$
    – JumpJump
    Nov 7, 2015 at 16:52
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    $\begingroup$ This is indeed classical, see e.g. the 1997 paper by Chambolle & P.L. Lions (and references therein). $\endgroup$ Nov 7, 2015 at 17:00
  • $\begingroup$ $A$ has to be defined as a set-valued map, not as a single-valued map as you did. Then it is maximal monotone, yes. $\endgroup$ Nov 9, 2015 at 14:47

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