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

将构建算法来计算在涉及非凸体能量泛函且没有经典极小值的变分中出现的问题的广义解。由参数化杨氏测度和（最小序列的）弱极限组成的广义解确实存在，并将通过这些算法进行数值计算。由于弱极限和杨氏测度通常是缓慢变化的量，因此与直接模拟振荡最小化序列相比，该方法具有相当大的经济性，可以很好地使用粗网格逼近。在许多物理问题中，观察到的宏观性质是精细量的平均值，只有通过光学显微镜或可能的电子显微镜才能看到。例如，许多常见的金属是由几种截然不同的金属组成的。通过对细层的直接平均来确定这种材料的性质的问题超出了当前计算机的能力范围。杨氏测量法非常简洁地描述了平均性质。在这个项目中，杨氏测度理论将被用于开发能够确定这些材料微观和宏观特性的计算机代码。

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.