The Dirichlet problem for the Jacobian equation in critical and supercritical Sobolev spaces.

We study existence and regularity of solutions to the Dirichlet problem for the prescribed Jacobian equation, det D u = f , where f is integrable and bounded away from zero. In particular, we take f ∈ L p , where p > 1 , or in L log L . We prove that for a Baire-generic f in either space there are no solutions with the expected regularity.