Energy Minimisers with Prescribed Jacobian

We consider the class of planar maps with Jacobian prescribed to be a fixed radially symmetric function f and which, moreover, fixes the boundary of a ball; we then study maps which minimise the 2p-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. Inst. H. Poincaré Anal. Non Linéaire 11(3):275-296, 1994). Some of our results also extend to the setting where the ball is replaced by $$ and boundary conditions are not prescribed.