Calculus of Variations and Geometric Measure Theory

A. Pratelli - E. Radici

On the planar minimal BV extension problem

created by pratelli on 18 Jul 2017
modified on 19 Oct 2020


Published Paper

Inserted: 18 jul 2017
Last Updated: 19 oct 2020

Journal: Rend. Lincei Mat. Appl.
Year: 2018


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.