Nonlinear elliptic boundary value problems

非线性椭圆边值问题

• 批准号: 0103823
• 负责人: Eun Heui Kim
• 金额: $ 6.9万
• 依托单位: University of Houston
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-07-15 至 2001-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0103823&HistoricalAwards=false
• 关键词: Nonlinear; elliptic; boundary; value; problems

NSF Award Abstract - DMS-0103823Mathematical Sciences: Nonlinear Elliptic Boundary Value ProblemsAbstractDMS-0103823KeyfitzThe major focus of this project is to develop systematic theories to understand the solution structures of nonlinear elliptic boundary value problems arising from change-of-type systems in multidimensional conservation laws, and from experiments in morphogenesis and ecological systems. It is a distinctive feature of multidimensional conservation laws in self-similar coordinates that they change type, being hyperbolic far from the origin but of mixed type near the origin. Analysis of behavior near the origin gives rise to interesting open problems in elliptic partial differential equations, including nonlinear free boundary value problems, degenerate oblique boundary problems, and quasilinear degenerate elliptic systems. Such problems arise in many different combinations in the study of two-dimensional Riemann problems, including those for the compressible Euler equations of gas dynamics. This project also investigates a class of singular non-quasimonotone elliptic systems that arise in models of biological morphogenesis experiments and certain predator-prey interactions. Multidimensional conservation laws are mathematical models for fundamental processes in physics and engineering, such as high-speed flows and supersonic jets. While tools for numerical simulations of multidimensional conservation laws have been developed extensively, there is very little analytical theory available. It is our goal to establish parts of the needed theory and to investigate the structure of solutions for such problems. The results of the project will have application to a wide a variety of physical systems.

NSF奖摘要- DMS-0103823数学科学：非线性椭圆边值问题摘要DMS-0103823 Keyfitz本项目的主要重点是发展系统的理论，以了解多维守恒律中的变型系统以及形态发生和生态系统实验中产生的非线性椭圆边值问题的解结构。 自相似坐标系中的多维守恒律的一个显著特征是它们改变类型，远离原点是双曲型，但靠近原点是混合型。 分析原点附近的行为会引起椭圆型偏微分方程中有趣的开放问题，包括非线性自由边值问题，退化斜边界问题和拟线性退化椭圆型系统。 这样的问题出现在许多不同的组合在研究二维黎曼问题，包括那些可压缩的气体动力学欧拉方程。 本计画也研究一类奇异非拟单调椭圆型系统，其出现在生物形态发生实验与某些捕食者-食饵互动的模型中。 多维守恒定律是物理学和工程学中基本过程的数学模型，如高速流动和超音速射流。 虽然多维守恒律的数值模拟工具已经得到了广泛的发展，但很少有可用的分析理论。 我们的目标是建立所需的理论的一部分，并调查这些问题的解决方案的结构。 该项目的结果将应用于各种各样的物理系统。