Variational problems and nonlinear equations from geometry

几何变分问题和非线性方程

• 批准号: 0800084
• 负责人: Matthew Gursky
• 金额: $ 15万
• 依托单位: University of Notre Dame
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-10-15 至 2012-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0800084&HistoricalAwards=false
• 关键词: Variational; problems; nonlinear; equations; geometry

The research activities supported by this award are in the field of geometric analysis, and the specific problems under investigation are located at the intersection of three fields: partial differential equations, differential geometry, and mathematical physics. For example, the functional determinant of an elliptic operator is a problem originating in spectral theory and mathematical physics, and the analysis of the particular problem we consider is a variational problem leading to a fourth order elliptic equation. The associated Lagrangian is unbounded, and the existence of solutions and their qualitative properties is highly nontrivial. Similar equations are used to model the properties of thin films. Another set of problems the principal investigator will explore involves certain fully nonlinear equations arising in conformal geometry and the theory of optimal transportation. Higher order regularity for these equations fails when the underlying manifold is negatively curved. While the structural reasons for this lack of regularity are in some sense understood (i.e., how the nonlinearity leads to the failure of certain estimates), there is no geometric description of the nature of singularities. The interaction of geometry and analysis dates back to at least the eighteenth century, and yet continues to be an important and highly active field of mathematical research. The classical subject of geometry grew out of our desire to understand certain properties of the physical world, and differential geometry was developed to understand the geometry of curved spaces--for example, the curvature of the surface of the earth, or the curvature of space by matter as predicted by general relativity. In the same way that Descartes realized that plane geometry can be studied using algebra, so differential geometry can be studied using techniques from analysis, especially differential equations. The research in this project involves disparate problems from geometry and mathematical physics, but its various aspects are united by the role played by mathematical analysis in their study.

该奖项支持的研究活动在几何分析领域，所研究的具体问题位于三个领域的交集：部分微分方程，差异几何和数学物理学。 例如，椭圆运算符的功能决定因素是源于光谱理论和数学物理学的问题，我们认为的特定问题的分析是导致第四阶椭圆方程的变异问题。 相关的Lagrangian是无限的，解决方案的存在及其定性特性是高度不平凡的。 类似的方程式用于对薄膜的性质进行建模。主要研究者将探索的另一组问题涉及在保形几何形状和最佳运输理论中产生的某些完全非线性方程。当底层歧管呈负弯曲时，这些方程式的高阶规则性会失败。尽管缺乏规律性的结构原因在某种意义上是理解的（即非线性如何导致某些估计的失败），但对奇异性本质没有几何描述。 几何和分析的相互作用至少可以追溯到18世纪，但仍然是数学研究的重要且高度活跃的领域。几何学的经典主题是我们希望理解物理世界某些特性的愿望，并开发了差异几何形状来理解弯曲空间的几何形状 - 例如，地球表面的曲率或通过一般相对性预测的空间曲率。就像笛卡尔意识到可以使用代数研究平面几何形状一样，可以使用分析中的技术（尤其是微分方程）研究差异几何形状。该项目中的研究涉及几何和数学物理学的不同问题，但其各个方面是由数学分析在他们的研究中所起的作用而团结的。