Mathematical Sciences: Partial Differential Equations and Applications

这项工作的重点是一类非线性偏微分方程产生的物理和工程模型。一组特殊的问题集中在自由边界问题上，这些问题发生在熔化和凝固过程、空腔和射流以及机械加工或电镀的电化学过程的研究中。在这类问题中，要寻找解的领域边界的一部分是未知的，必须与解同时找到。在这类问题上已经取得了实质性的进展，但更新和更现实的模型导致了复杂的方程，特别是当从二维到三维描述时。涉及自由边界的优化设计工作也将进行。这些问题通常可以表述为控制函数的最小化问题，其解可能不可微。这种类型的问题发生，例如，在试图减少大坝的压力。控制变量要么出现在数据中，要么出现在所得微分方程的系数中。自由边界条件可能出现在作为控制变量的曲面上。最近关于构造具有一定折射特性的光学透镜的工作简化为求解两个非线性泛函微分方程的系统。描述了一个具有一个对称轴的全局透镜的构造，它将来自两个光源的所有光线聚焦到两个给定的收集点上，这些收集点都在对称轴上，尽管不清楚当不是所有的点都在对称轴上时会发生什么。未来的工作计划在这个方程上，应用于卫星天线建模。

This work focuses on a large class of nonlinear partial differential equations arising from physical and engineering models. A particular set of problems centers on free boundary problems occurring in the study of melting and solidification processes, cavity and jet flows and electrochemical processes from machining or electroplating. In such problems, a part of the boundary of the domain where the solution is to be found is unknown and must be found simultaneously with the solution. Substantial progress has been made on problems of this type but newer and more realistic models lead to complex equations, especially when going from two to three-dimensional descriptions. Work on optimal design involving free boundaries will also be carried out. These can often be formulated as a minimization problem for some function of a control in which the solution may not be differentiable. Problems of this type occur, for example, in trying to minimize the pressure in a dam. The control variables appear either in the data or in the coefficients of the resulting differential equation. The free boundary conditions may occur on a surface which is the control variable. Recent work on construction for an optical lens with certain refraction properties reduces to solving systems of two nonlinear functional differential equations. The construction of a global lens with one axis of symmetry which focuses all rays from two sources into two given collecting points all on the axis has been described although it is not clear what happens when not all points are on the axis of symmetry. Future work is planned on this equation where applications to satellite antenna modeling are expected.