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On the equation \({{\rm det}\,\nabla{u}=f}\) with no sign hypothesis

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Abstract

We prove existence of \({u\in C^{k}(\overline{\Omega};\mathbb{R}^{n})}\) satisfying

$$\left\{\begin{array}{ll} det\nabla u(x) =f(x) \, x\in \Omega\\ u(x) =x \quad\quad\quad\quad x\in\partial\Omega\end{array}\right.$$

where k ≥ 1 is an integer, \({\Omega}\) is a bounded smooth domain and \({f\in C^{k}(\overline{\Omega}) }\) satisfies

$$\int\limits_{\Omega}f(x) dx={\rm meas} \Omega$$

with no sign hypothesis on f.

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Authors and Affiliations

  1. Section de Mathématiques, EPFL, 1015, Lausanne, Switzerland

    G. Cupini, Bernard Dacorogna & O. Kneuss

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  1. G. Cupini
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  2. Bernard Dacorogna
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  3. O. Kneuss
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Correspondence to Bernard Dacorogna.

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Cupini, G., Dacorogna, B. & Kneuss, O. On the equation \({{\rm det}\,\nabla{u}=f}\) with no sign hypothesis. Calc. Var. 36, 251–283 (2009). https://doi.org/10.1007/s00526-009-0228-3

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  • DOI: https://doi.org/10.1007/s00526-009-0228-3

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