Mathematical Sciences: Calculation of Young Measure Valued Solutions Arising in the Calculus of Variations

将构造算法来计算广义 变分法中出现的问题的解决方案， 涉及非凸体能泛函，且不 经典极小 广义解，由一个 参数化Young测度与极小化的弱极限 序列）确实存在，并将通过这些数值计算 算法 建议的方法提供了相当大的经济 振荡极小化序列的超直接模拟 由于弱极限和Young测度是缓慢变化的 一般来说，可以很好地近似使用 粗网格 在许多物理问题中， 属性是仅可见的细尺度数量的平均值 通过光学或电子显微镜。 比如说， 许多普通金属由几层不同的 不同的金属成分。 的确定问题 这种材料的性能通过直接平均超过罚款 这超出了当前计算机的范围。 年轻 测量非常简洁地表征了平均属性。 在这个项目中，Young测度理论将被用来 开发计算机代码，能够确定 这些材料的微观和宏观性质。

Algorithms will be constructed to calculate generalized solutions of problems arising in the calculus of variations that involve non-convex bulk energy functionals and which have no classical minima. The generalized solutions, consisting of a parameterized Young measure and a weak limit (of a minimizing sequence) do exist and will be calculated numerically by these algorithms. The proposed approach offers considerable economy over direct simulation of the oscillatory minimizing sequences since the weak limit and Young measure are slowly varying quantities in general, which may be well approximated using coarse meshes. In many physical problems, the observed macroscopic properties are averages of fine-scale quantities visible only through optical or possibly electron microscopes. For example, many common metals consist of layers of several distinctly different component metals. The problem of determining the properties of such a materials by direct averaging over the fine layers is beyond the scope of current computers. The Young measure characterizes the averaged properties very succinctly. In this project, the theory of Young measures will be utilized to develop computer codes capable of determining both the microscopic and macroscopic properties of such materials.