On the planar minimal BV extension problem

Given a continuous, injective function j defined on the boundary of a planar open set W, we consider the problem of minimizing the total variation among all the BV homeomorphisms on W coinciding with j on the boundary. We find the explicit value of this infimum in the model case when W is a rectangle. We also present two important consequences of this result: first, whatever the domain W is, the infimum above remains the same also if one restricts himself to consider only W1;1homeomorphisms. Second, any BV homeomorphism can be approximated in the strict BV sense with piecewise affine homeomorphisms and with diffeomorphisms.