Some results on the Jacobian equation with integrable data and a nonlinear open mapping principle

I will present results on the Jacobian equation det Du = f where f\in L^p. In particular, I will present results on generic non-existence of solutions on bounded domains as well as results on symmetry properties of the solution answering questions by Ye, Hogan-McIntosh-Zhang and Hélein. An important tool will be a nonlinear open mapping principle that is of independent interest. In fact, as an application of this principle I will show that the set of initial data for which there are dissipative weak solutions in L^p_t L^2_x of the incompressible Euler equations is meagre in the space of solenoidal L^2 fields.