轴对称情形下neo-Hookean能量极小值的存在性
On the existence of minimizers for the neo-Hookean energy in the axisymmetric setting
摘要：
设$Ω$是$\R^3$中的光滑有界轴对称集.本文研究了一类轴对称映射的neo-Hookean能量极小元的存在性。由于能量中临界指数的出现，我们必须面对一个缺乏紧性的问题。事实上，如Conti-De Lellis的例子所示，能量集中的现象可以发生，从而阻止最小化序列的$W^{1,2}(Ω,\mathbb{R}^3)$的强收敛以及该序列的余因子的等可积性。我们证明了这种现象只能发生在磁畴的对称轴上。因此，如果我们考虑不包含对称轴的域，那么最小化器确实存在。我们还提供了一个部分描述的缺乏紧凑性的笛卡尔电流。然后，我们研究的情况下，$Ω$不一定是轴对称的，但边界数据是仿射的。在这种情况下，如果我们不允许空化（也不在内部，也不在边界），那么仿射扩张是唯一的极小化，也就是说，二次多凸能量是$W^{1,2}$ -拟凸在我们的容许空间。最后，在轴对称区域不含对称轴的情况下，我们首次得到了$3$ D neo-Hookean材料的能量-动量方程弱解的存在性.
英文摘要：
Let $Ω$ be a smooth bounded axisymmetric set in $\R^3$. In this paper we investigate the existence of minimizers of the so-called neo-Hookean energy among a class of axisymmetric maps. Due to the appearance of a critical exponent in the energy we must face a problem of lack of compactness. Indeed as shown by an example of Conti-De Lellis, a phenomenon of concentration of energy can occur preventing the strong convergence in $W^{1,2}(Ω,\mathbb{R}^3)$ of a minimizing sequence along with the equi-integrability of the cofactors of that sequence. We prove that this phenomenon can only take place on the axis of symmetry of the domain. Thus if we consider domains that do not contain the axis of symmetry then minimizers do exist. We also provide a partial description of the lack of compactness in terms of Cartesian currents. Then we study the case where $Ω$ is not necessarily axisymmetric but the boundary data is affine. In that case if we do not allow cavitation (nor in the interior neither at the boundary) then the affine extension is the unique minimizer, that is, quadratic polyconvex energies are $W^{1,2}$-quasiconvex in our admissible space. At last, in the case of an axisymmetric domain not containing its symmetry axis, we obtain for the first time the existence of weak solutions of the energy-momentum equations for $3$D neo-Hookean materials.
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