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arXiv · 2012.10132

Energy minimisers with prescribed Jacobian

André Guerra·Lukas Koch·Sauli Lindberg
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Abstract

We study the symmetry and uniqueness of maps which minimise the $np$-Dirichlet energy, under the constraint that their Jacobian is a given radially symmetric function $f$. We find a condition on $f$ which ensures that the minimisers are symmetric and unique. In the absence of this condition we construct an explicit $f$ for which there are uncountably many distinct energy minimisers, none of which is symmetric. Even if we prescribe the maps to be the identity on the boundary of a ball we show that the minimisers need not be symmetric. This gives a negative answer to a question of Hélein (Ann. Inst. H. Poincaré Anal. Non Linéaire 11 (1994), no. 3, 275-296).

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André Guerra, Lukas Koch, Sauli Lindberg. 2020-12-18. Energy minimisers with prescribed Jacobian. https://doi.org/10.1007/s00205-021-01699-4

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