Mathematical Sciences: Nonlinear Partial Differential Equations and Nonconvex Variational Problems

该项目的大部分工作仍在继续 研究数学分析问题， 通常涉及一些特定的应用或物理 模型 有四种类型的研究，第一 关注非凸的奇异扰动 变分问题 这意味着定位功能 最小化某个非凸变换的积分 一个函数类。 解决方案通常不是 独特. 为了挑选出更合理的解决方案， 引入扰动以惩罚 在分段常数解之间出现的界面。 这种对问题的适应导致了一种形式的“偏好” 在某些情况下已经确定的解决方案。 那里 还有许多其他重要的病例没有得到治疗; 两个特别的例子是未知函数 是梯度，未知数是矩阵的情况下， 值函数。 将在约束反作用方面做更多的工作- 扩散过程，分析 非线性抛物方程的解 战线 通常代表相边界，它们的速度通常 取决于它们的局部平均曲率，例如发生在 金属中晶界的运动。 所述问题被 这里分析的是受质量影响的梯度流 约束。 战线会迅速发展， 以曲率相关的法向速度传播。 的 质量约束将被抑制， 传播规则仍有待建立。 在优化设计中出现的非凸问题中， 寻求最大限度地降低材料成本，同时保持 必要的力量。 在数学上， 最小化问题是一个约束最小梯度问题 问题. 构建解决方案的处方 Kohn和斯特朗给出了二维 案子 这项工作的目的是发展一个严格的分析， 这种方法以及寻找一种方法来分析 更高维度的问题。 工作也将做的分支现象， 孤独的波浪 1954年，弗里德里希斯和海尔斯建立了 水波方程分支解的存在性 对于足够小的速度和振幅。 还没有 建立是否二级分叉的 可以出现第一分支。 本研究的对象将是 为了证明这种现象是不可能的， 保证解的唯一性的条件， 给定振幅或速度。

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.