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
$Du=JDu$, $u_{\partial \Omega}=f$
has a solution in a reasonable space ($BV(\Omega)$ for instance)?
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 $Du=JDu$, $u_{\partial \Omega}=f$ has a solution in a reasonable space ($BV(\Omega)$ for instance)? 

