Mathematical Sciences: Nonlinear Partial Differential Equations and Nonconvex Variational Problems

数学科学：非线性偏微分方程和非凸变分问题

• 批准号: 8901726
• 负责人: Peter Sternberg
• 金额: $ 3.2万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1989
• 资助国家: 美国
• 起止时间: 1989-06-15 至 1992-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8901726&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Nonlinear; Partial; Differential

Much of the work in this project represents continuing investigations into problems of mathematical analysis, generally related to some particular application or physical model. There are four categories of study, the first concerned with singular perturbations of nonconvex variational problems. By this one means locating functions which minimize an integral of some nonconvex transformation of a class of functions. Solutions are generally not unique. To single out more reasonable solutions, singular perturbations are introduced to penalize the formation of interfaces which occur among piecewise constant solutions. This adaptation of the problem leads to a form of "preferred" solutions which have been identified in certain cases. There are many other important cases which have not been treated; two in particular are the cases where the unknown functions are gradients and the case where the unknowns are matrix- valued functions. Additional work will be done on constrained reaction- diffusion processes, analyzing the front propagation of solutions to nonlinear parabolic equations. The fronts typically represent phase boundaries and their velocity often depends on their local mean curvature such as occur in the motion of grain boundaries in metals. The problem to be analyzed here is a gradient flow subject to a mass constraint. The fronts will develop rapidly and then propagate with a curvature-dependent normal velocity. The mass constraint will be inhibiting so that the actual form of the propagation rule remains to be established. In a nonconvex problem arising in optimal design, one seeks to minimize the cost of material while maintaining the necessary strength. Mathematically, the form of the minimization problems is that of a constrained least gradient problem. A prescription for the construction of a solution has been given by Kohn and Strang for the two-dimensional case. This work aims at developing a rigorous analysis of this method as well as looking for a means to analyze the question in higher dimensions. Work will also be done on branching phenomena of solitary waves. In 1954, Friedrichs and Hyers established the existence of branch solutions of the water wave equation for sufficiently small speed and amplitude. It has not been established whether or not secondary bifurcations of the first branches can occur. The object of this study will be to show that this phenomenon is not possible and to establish conditions which guarantee the uniqueness of solutions having given amplitude or speed.

该项目中的许多工作代表着对数学分析问题的持续调查，通常与某些特定的应用或物理模型有关。 研究有四种类别，第一个与非凸变体问题的奇异扰动有关。 通过这种方式，请定位函数，以最大程度地减少一类函数的某些非凸转换的积分。 解决方案通常不是唯一的。 为了挑出更合理的解决方案，引入了奇异的扰动，以惩罚分段恒定解决方案之间发生的接口的形成。 该问题的这种适应导致了在某些情况下已确定的“首选”解决方案的一种形式。 还有许多其他重要案例尚未得到治疗； 尤其是两个是未知函数是梯度的情况，而未知数是基质值函数的情况。 将在约束反应扩散过程上进行其他工作，分析非线性抛物线方程溶液的前端传播。 前沿通常表示相位边界，它们的速度通常取决于其局部平均曲率，例如在金属中晶界运动中发生。 这里要分析的问题是受质量约束的梯度流。 前部将迅速发展，然后以曲率依赖性的正常速度传播。 质量约束将被抑制，以便尚待建立传播规则的实际形式。 在最佳设计中引起的非凸问题中，人们试图在保持必要的强度的同时最大程度地降低材料成本。 从数学上讲，最小化问题的形式是梯度最小的问题。 Kohn和Strang为二维情况提供了建设解决方案的处方。 这项工作旨在对这种方法进行严格的分析，并寻找一种在更高维度中分析问题的方法。 还将在孤立波的分支现象上完成工作。 1954年，弗里德里奇（Friedrichs and Hyers）建立了水波方程的分支解决方案，以实现足够小的速度和振幅。 尚未确定是否可能发生第一分支的次要分叉。 这项研究的目的是表明这种现象是不可能的，并建立了保证具有振幅或速度的解决方案独特性的条件。