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

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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.