Let $E$ have finite perimeter in $\mathbb R^n$. Consider the minimization of

$$P(B)-\int_{\mathbb R^n\setminus L}|D\chi_E|$$

among the sets $B$ of finite perimeter differing from $E$ only inside $L$ (the negative piece of the minimized quantity is not influent, since it does not depend on $B$, but it's there to stress the idea that we care just about the perimeter inside $L$). Let $\bar E$ be the minimizer, and that the minimum is not zero. Suppose also that $L$ has finite perimeter.

I'm looking for a geometrically inspired way for defining a function $\Psi$ which sends a piece (I would like to ask "all" but it might be unfeasible) of $\mathcal FL\cap E$ to $\mathcal F\bar E\cap L$ bijectively, is $H^{n-1}$-measurable and decreases (more precisely, does not increase) $(n-1)$-dimensional area.

The idea should be that $\Psi$ be some kind of projection.

In the case that $L$ is a ball and $E$ is a thin rectangle around its diameter it is clear that usual projections don't quite work.

On the other hand, such a map surely exists, by the consideration that the area of the target is smaller than the one of the domain, and both of them are rectifiable.

My question is if there is a famous geometric/constructive way of doing it, perhaps using some minimization principle, a flow, etc., or under more restrictive hypotheses on $L$.