JournalsrlmVol. 29, No. 3pp. 511–555

On the planar minimal BV extension problem

  • Aldo Pratelli

    Universität Erlangen-Nürnberg, Germany
  • Emanuela Radici

    Università degli Studi dell'Aquila, Italy
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Abstract

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

Cite this article

Aldo Pratelli, Emanuela Radici, On the planar minimal BV extension problem. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 29 (2018), no. 3, pp. 511–555

DOI 10.4171/RLM/819