Mathematical Sciences: Semilinear Elliptic Equations and Systems

这个项目的工作结合了几何激励的问题集在一个分析框架。它代表着四个领域研究的继续。在第一章中，我们将继续研究完备流形上具有指定标量曲率或高斯曲率的共形度量的存在性和刻画性质。这等价于在流形上寻找椭圆非线性方程的正解，其线性项是度量中的Laplace-Beltrami算子，其标量曲率表现为未知函数的乘子。虽然人们对紧凑型机箱的了解很多，但在非紧凑型机箱中所做的工作相对较少，直到最近。第二条线的研究将集中于发展对天体物理学中Matukuma方程的有限质量解和无限质量解的完整理解。这些是1930年首次提出的半线性方程，用来模拟全球星团的动力学。这些方程被认为是爱丁顿1915年模型的改进。这里的未知函数是引力势，对于有限总质量的情况，直到最近才被证明有正的整体解(相应的爱丁顿方程没有)。关于这两种模式及其解决方案之间的差异，仍有许多问题没有得到回答。还将在确定天体物理学的Lane-Emden方程是否存在振动解和非振动解以及分析奇异径向解的渐近行为方面进行工作。此外，最近关于趋化性的工作导致了关于抛物线系统解的新结果，该结果描述了细胞对环境中的化学物质的定向运动。在确定细胞如何向浓度较高的地方移动，然后聚集方面，还有进一步的工作要做。

Work on this project combines geometrically motivated problems set in an analytic framework. It represents a continuation of research in four areas. In the first, work will continue in establishing existence and describing properties of conformal metrics with prescribed scalar or Gaussian curvature on complete manifolds. This is equivalent to finding positive solutions of an elliptic, nonlinear equation on the manifold, whose linear term is the Laplace-Beltrami operator in the metric with scalar curvature appearing as a multiplier of the unknown function. Although much is known about the compact case, relatively little has been done in the non-compact setting until recently. A second line of investigation will focus on developing a complete understanding of finite mass solutions and infinite mass solutions of the Matukuma equations in astrophysics. These are semilinear equations, first proposed in 1930, to model the dynamics of global star clusters. These equations are believed to be an improvement of Eddington's 1915 model. The unknown function here is the gravitational potential which, for the finite total mass case, has only recently been shown to have positive entire solutions (the corresponding Eddington equation has none). Many questions remain unanswered regarding the two models and discrepancies observed between their solutions. Work will also be done in determining whether the Lane- Emden equation of astrophysics has oscillatory as well as non- oscillatory solutions and analyzing the asymptotic behavior of singular radial solutions. In addition, relatively recent work on chemotaxis has led to new results on solutions of parabolic systems describing the oriented movement of cells in response to chemicals in the environment. Further work remains in determining how the cells move toward places of higher concentration and then aggregate.