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

  • G. Cupini
  • Bernard DacorognaEmail author
  • O. Kneuss


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.

Mathematics Subject Classification (2000)



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© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Section de MathématiquesEPFLLausanneSwitzerland

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