Nonlinear elliptic boundary value problems

NSF奖摘要-DMS-0103823数学科学：非线性椭圆边值问题摘要DMS-0103823Keyfitz本项目的主要重点是发展系统理论来理解由多维守恒律中的类型变化系统引起的非线性椭圆边值问题的解结构，以及从形态发生和生态系统的实验中得到的。自相似坐标下的多维守恒定律的一个显著特点是变型，远离原点呈双曲线，而在原点附近呈混合型。对原点附近的行为的分析引出了椭圆型偏微分方程组中有趣的公开问题，包括非线性自由边值问题、退化斜边值问题和拟线性退化椭圆组。在二维黎曼问题的研究中，这类问题出现在许多不同的组合中，包括气体动力学的可压缩欧拉方程。本项目还研究了一类奇异的非拟单调椭圆型系统，该系统存在于生物形态发生实验和某些捕食者-食饵相互作用的模型中。多维守恒定律是物理和工程中基本过程的数学模型，如高速流动和超音速喷流。虽然多维守恒定律的数值模拟工具已经得到了广泛的开发，但可用的分析理论很少。我们的目标是建立所需理论的一部分，并调查此类问题的解决方案的结构。该项目的成果将广泛应用于各种物理系统。

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.