On the planar minimal BV extension problem

Abstract

Given a continuous, injective function $\phi$ 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 $\phi$ 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.