Differential Inclusions and Young Measures Involving Prescribed Jacobians
Differential Inclusions and Young Measures Involving Prescribed Jacobians
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涉及规定雅可比行列式的微分包含和年轻测度
DOI：
10.1137/140968860
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发表时间：
2015
影响因子：
2
通讯作者：
Koumatos K
中科院分区：
文献类型：
--
作者：
Koumatos K
本文从凸积分的思想出发，给出了一种刻画小于空间维的映射的梯度所生成的Young测度的方法，其雅可比行列式受一系列约束。在弹性力学和流体力学理论中，有两种特殊情况特别重要：当(A)生成的梯度具有一致地远离零的正的雅可比时，以及(B)基础变形是不可压缩的，对应于它们的雅可比行列式始终为1。这一刻画结果，以及它的各种推论，强调了次临界Soblev空间中雅可比行列式的灵活性，并给出了对已知的点状雅可比病理的更系统和更一般的观点。最后，我们证明了对于小于该维的非线性弹性力学，-拟凸和-保向拟凸都是不适合的凸性条件，其中假定能量随着雅可比趋于零时爆炸。
This work presents a general principle, in the spirit of convex integration, leading to a method for the characterization of Young measures generated by gradients of maps inwithless than the space dimension, whose Jacobian determinant is subjected to a range of constraints. Two special cases are particularly important in the theories of elasticity and fluid dynamics: when (a) the generating gradients have positive Jacobians that are uniformly bounded away from zero and (b) the underlying deformations are incompressible, corresponding to their Jacobian determinants being constantly one. This characterization result, along with its various corollaries, underlines the flexibility of the Jacobian determinant in subcritical Sobolev spaces and gives a more systematic and general perspective on previously known pathologies of the pointwise Jacobian. Finally, we show that, forless than the dimension,-quasi-convexity and-orientation-preserving quasi-convexity are both unsuitable convexity conditions for nonlinear elasticity where the energy is assumed to blow up as the Jacobian approaches zero.