Resolutions of the prescribed volume form equation. (English) Zbl 0857.35025

Summary: For a given volume form \(f dx\) on a bounded regular domain \(\Omega\) in \(\mathbb{R}^n\), we are looking for a transformation \(u\) of \(\Omega\), keeping the boundary fixed and which sends the Lebesgue measure \(dx\) into \(f dx\) (i.e. we solve \(\text{det}(\nabla u)=f\)). For \(f\) in various spaces, we propose two different constructions which ensure the existence of \(u\) with some gain of regularity. Our methods permit the recovery Dacorogna and Moser’s results [B. Dacorogna and J. Moser, Ann. Inst. H. Poincaré, Anal. Nonlin. 7, No. 1, 1-26 (1990; Zbl 0707.35041)], but also, we prove the existence of such \(u\) in Hölder spaces for \(f\) in \(C^0\), or even in \(L^\infty\).


35F30 Boundary value problems for nonlinear first-order PDEs


Zbl 0707.35041
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