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We study the symmetry and uniqueness of maps which minimise the$np$-Dirichlet energy, under the constraint that their Jacobian is a givenradially symmetric function $f$. We find a condition on $f$ which ensures thatthe minimisers are symmetric and unique. In the absence of this condition weconstruct an explicit $f$ for which there are uncountably many distinct energyminimisers, none of which is symmetric. Even if we prescribe the maps to be theidentity on the boundary of a ball we show that the minimisers need not besymmetric. This gives a negative answer to a question of H\'{e}lein (Ann. Inst.H. Poincar\'{e} Anal. Non Lin\'{e}aire 11 (1994), no. 3, 275-296).

Energy minimisers with prescribed Jacobian