Mathematical Sciences: Partial Regularity in the Calculus ofVariations

数学科学：变分演算中的部分正则性

• 批准号: 8704111
• 负责人: Ronald Gariepy
• 金额: $ 2.93万
• 依托单位: University of Kentucky Research Foundation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1987
• 资助国家: 美国
• 起止时间: 1987-07-15 至 1990-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8704111&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Partial; Regularity; Calculus

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.

工作计划在变分法的问题。 具体而言，主要研究者关注的是 整函数极小值的正则性和存在性 其被积函数表现出某种奇异行为。 存在论的基础是C. B.莫雷， 分离出拟凸性的概念， 为了生存 由L.C.埃文斯 严格拟凸性保证了 最小化者 最近的文献指出，在研究中， 非线性弹性力学平衡边值问题 保证规律性所需的生长条件是 过于限制，无法包括超弹性的真实模型 功能最小化的材料（即其 积分）对应于存储的能量。 这个问题发生在 特别是当处理可压缩材料时。 我们将对表现出无界性的被积函数进行研究， 只受限制，即行列式的 容许函数的梯度（代表流体静力学 压力是积极的。 极小值的存在性不是问题 在这种情况下，规律性是。 变分问题 产生了一个欧拉方程，但并不清楚 最小值甚至是这个方程的弱解。 迄今工作 然而，这表明，仔细分析可受理的 变化将产生期望的规律性。 这项研究有望在研究中得到应用 非线性弹性和不可压缩的流体流动。