论文标题
用规定的雅各布的能量最小化器
Energy minimisers with prescribed Jacobian
论文作者
论文摘要
我们研究了地图的对称性和独特性，这些地图最小化$ np $ -dirichlet Energy的限制是在限制的雅各布是给定的径向对称函数$ f $的限制下。我们找到了$ f $的条件，可确保最小化器对称和独特。在没有这种情况的情况下，我们构建了一个明确的$ f $，对于无数截然不同的能量最小剂，它们都不是对称的。即使我们规定了地图是球边界上的身份，我们也表明，最小化器不必是对称的。这给出了Hélein问题的负面答案（Ann。H.PoincaréAnal。NonLinéaire11（1994），第3、275-296号）。
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).
