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Let $\Omega$ be a bounded region in $R^n$ and $J\in (L^2(\Omega))^n$ with $|J| \leq 1$ a.e. in $\Omega$. Under what conditions the equation

$Du=J|Du|$, $u|_{\partial \Omega}=f$

has a solution in a reasonable space ($BV(\Omega)$ for instance)?

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Is it intended that $|J|$ could be $<1$ and, in this case, $Du$ is forced to equal $0$? –  Will Sawin Oct 18 '13 at 17:34
    
Yes. But of course one may want to impose certain conditions on $J$, $|J|=1$ for instance, to be able to solve the equation. –  Tom Oct 18 '13 at 17:50
    
In the case that everything is smooth and $|J| = 1$, it seems to me that the solution $u$ is any function constant on the hypersurfaces orthogonal to the flow lines of $J$; the values of $u$ can be any function along a line transversal to these hypersurfaces. –  Connor Mooney Oct 18 '13 at 19:03
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