Mathematical Sciences: Partial Regularity in the Calculus ofVariations

工作计划在变分演算的问题上进行。具体地说，主要研究者关注的是积分函数的最小值的规律性和存在性，其被积项表现出某种类型的奇异行为。存在理论的基础是C.B. Morrey的结果，他将准自凸性的概念作为存在的基本成分。L.C. Evans提出了一个更强的性质，称为严格拟自凸性，它保证了极小值的部分正则性。最近的文献指出，在非线性弹性平衡边值问题的研究中，保证规则性所需的生长条件过于严格，不能包括超弹性材料的现实模型，其中函数被最小化（即其积分）对应于存储的能量。当处理可压缩材料时，这个问题尤其突出。工作将用表现出无界性的积分来完成，仅受允许函数（代表静水压力）梯度的行列式为正的限制。在这种情况下，最小化的存在不是问题，但规律性是问题。变分问题产生了一个欧拉方程，但我们甚至不清楚最小化是否是这个方程的弱解。然而，迄今为止的工作表明，对可接受的变化进行仔细分析将产生期望的规律性。该研究有望在非线性弹性和不可压缩流体流动的研究中得到应用。

Work is planned on problems in the calculus of variations. Specifically, the principal investigator is concerned with the regularity and existence of minimizers of integral functions whose integrands exhibit a certain type of singular behavior. Fundamental to existence theory is a result of C.B. Morrey which isolates the notion of quasiconvexity as an essential ingredient for existence. A stronger property formulated by L.C. Evans called strict quasiconvexity guarantees partial regularity of minimizers. It has been pointed out in recent literature that in studies of equilibrium boundary value problems in nonlinear elasticity that the growth conditions required to guarantee regularity is too restrictive to include realistic models for hyperelastic materials where the functional being minimized (i.e. its integral) corresponds to stored energy. This problem occurs in particular when one is dealing with compressible materials. Work will be done with integrands exhibiting unboundedness, subject only to the restriction that the determinant of the gradient of admissible functions (representing hydrostatic pressure) be positive. Existence of minimizers is not a problem in this context but regularity is. The variational problem yields an Euler equation but it is not at all clear that a minimizer is even a weak solution of this equation. Work to date suggests nevertheless that a careful analysis of admissible variations will yield the desired regularity. This research is expected to find applications in the study of nonlinear elasticity and incompressible fluid flow.