## 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 $$L^\infty$$.

### MSC:

 35F30 Boundary value problems for nonlinear first-order PDEs

### Keywords:

Jacobian determinant equation

Zbl 0707.35041
Full Text:

### References:

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