Nonlinear elliptic boundary value problems

数学科学：非线性椭圆边值问题摘要本项目的主要目的是建立系统的理论，以理解多维守恒律下系统类型变化引起的非线性椭圆边值问题的解结构，以及形态发生和生态系统的实验。自相似坐标下多维守恒律的一个显著特征是类型变化，远离原点时为双曲型，靠近原点时为混合型。椭圆型偏微分方程在原点附近的行为分析引起了一些有趣的开放问题，包括非线性自由边值问题、退化斜边问题和拟线性退化椭圆系统。这类问题在二维黎曼问题的研究中以许多不同的组合出现，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非拟单调椭圆系统，这些系统出现在生物形态发生实验和某些捕食者-猎物相互作用的模型中。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射流。虽然多维守恒定律的数值模拟工具已经得到了广泛的发展，但可用的解析理论很少。我们的目标是建立所需理论的一部分，并研究这些问题的解决方案的结构。该项目的结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.