Calculus of Variations and Geometric Measure Theory
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A. Pratelli - E. Radici

On the planar minimal BV extension problem

created by pratelli on 18 Jul 2017
modified on 01 Sep 2017



Inserted: 18 jul 2017
Last Updated: 1 sep 2017

Year: 2017


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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