Calculus of Variations and Geometric Measure Theory

G. Cupini - B. Dacorogna - O. Kneuss

On the equation $\det \,Du=f$ with no sign hypothesis

created by cupini on 26 Jan 2009
modified on 14 Jan 2010


Published Paper

Inserted: 26 jan 2009
Last Updated: 14 jan 2010

Journal: Calc. Var. Partial Differential Equations
Volume: 36
Pages: 251-283
Year: 2009


We prove existence of $u\in C^{k}\left( \overline{\Omega};\mathbb{R}^n\right)$ satisfying \[ \left\{ \begin{array} [c]{cl} \det\nabla u\left( x\right) =f\left( x\right) \smallskip & x\in \Omega

u\left( x\right) =x & x\in\partial\Omega \end{array} \right. \] where $k\ge 1$ is an integer, $\Omega$ is a bounded smooth domain and $f\in C^{k}\left( \overline{\Omega}\right) $ satisfies \[ \int_{\Omega}f(x) dx=\textrm{meas}\,\Omega \] with no sign hypothesis on $f$.