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Resolutions of the prescribed volume form equation

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Abstract

For a given volume formfdx on a bounded regular domain Ω in IRn, we are looking for a transformationu of Ω, keeping the boundary fixed and which sends the Lebesgue measuredx intofdx (i.e. we solve det (Δu)=f. Forf in various spaces, we propose two different constructions which ensure the existence ofu with some gain of regularity. Our methods permit the recovery Dacorogna and Moser's results [4], but also, we prove the existence of suchu in Hölder spaces forf inC 0, or even inL ∞.

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Authors and Affiliations

  1. URA 1611 Ecole Normale Supérieure de Cachan, Centre de Mathématiques et de Leurs Applications CNRS, 61 Avenue du Président Wilson, 94235, Cachan Cedex, France

    Tristan Rivière & Dong Ye

  2. Bâtiment I, Département de Mathématiques, Université de Cergy-Pontoise, 8 le Campus, 95033, Cergy-Pontoise, France

    Dong Ye

Authors
  1. Tristan Rivière
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  2. Dong Ye
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Rivière, T., Ye, D. Resolutions of the prescribed volume form equation. NoDEA 3, 323–369 (1996). https://doi.org/10.1007/BF01194070

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  • DOI: https://doi.org/10.1007/BF01194070

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