Energy Minimisers with Prescribed Jacobian.

We consider the class of planar maps with Jacobian prescribed to be a fixed radially symmetric function and which, moreover, fixes the boundary of a ball; we then study maps which minimise the 2-Dirichlet energy in this class. We find a quantity which controls the symmetry, uniqueness and regularity of minimisers: if then minimisers are symmetric and unique; if is large but finite then there may be uncountably many minimisers, none of which is symmetric, although all of them have optimal regularity; if is infinite then generically minimisers have lower regularity. In particular, this result gives a negative answer to a question of Hélein (Ann.

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, , where is integrable and bounded away from zero. In particular, we take , where , or in . We prove that for a Baire-generic in either space there are no solutions with the expected regularity.