*Submitted Paper*

**Inserted:** 18 jul 2017

**Last Updated:** 30 jan 2018

**Year:** 2017

**Abstract:**

Given a continuous, injective function $\varphi$ defined on the boundary of a planar open set $\Omega$, we consider the problem of minimizing the total variation among all the $BV$ homeomorphisms on $\Omega$ coinciding with $\varphi$ on the boundary. We find the explicit value of this infimum in the model case when $\Omega$ is a rectangle. We also present two important consequences of this result: first, whatever the domain $\Omega$ is, the infimum above remains the same also if one restricts himself to consider only $W^{1,1}$ homeomorphisms. Second, any $BV$ homeomorphism can be approximated in the strict $BV$ sense with piecewise affine homeomorphisms and with diffeomorphisms.

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