Mathematical Sciences: Semilinear Elliptic Equations and Systems

这个项目的工作将几何驱动的问题集结合在一个分析框架中。它代表了四个领域研究的延续。首先，工作将继续在完全流形上建立具有规定标量曲率或高斯曲率的共形度量的存在性和描述性质。这相当于在流形上寻找一个椭圆型非线性方程的正解，其线性项是度量中的拉普拉斯-贝尔特拉米算子，标量曲率作为未知函数的乘数出现。虽然很多人都知道紧凑的情况下，相对较少已经做了在非紧凑的设置，直到最近。第二项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。这些方程被认为是对爱丁顿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，它直到最近才被证明有正的全解（相应的爱丁顿方程没有）。关于这两个模型和在它们的解决方案之间观察到的差异，许多问题仍未得到解答。在确定天体物理学的Lane- Emden方程是否具有振荡解和非振荡解以及分析奇异径向解的渐近行为方面也将进行工作。此外，最近在趋化性方面的工作已经导致了抛物线系统解决方案的新结果，描述了细胞对环境中化学物质的定向运动。进一步的工作仍在确定细胞如何向高浓度的地方移动，然后聚集。

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.