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

Cupini, G. ; Dacorogna, Bernard ; Kneuss, O.

In: Calculus of Variations and Partial Differential Equations, 2009, vol. 36, no. 2, p. 251-283

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    Title
    • On the equation $${{\rm det}\,\nabla{u}=f}$$ with no sign hypothesis
    Author
    • Cupini, G.. Section de Mathématiques, EPFL, 1015, Lausanne, Switzerland
    • Dacorogna, Bernard. Section de Mathématiques, EPFL, 1015, Lausanne, Switzerland
    • Kneuss, O.. Section de Mathématiques, EPFL, 1015, Lausanne, Switzerland
    Document Type
    • Postprint
    Language
    • English
    Published in
    • Calculus of Variations and Partial Differential Equations, 2009, vol. 36, no. 2, p. 251-283. Springer-Verlag
    Other Version
    Classification
    • Mathematics
    OAI-PMH Identifier
    • oai:doc.rero.ch:321100
    Summary
    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