Variational problems and nonlinear equations from geometry

该奖项支持的研究活动是在几何分析领域，所调查的具体问题位于三个领域的交叉点：偏微分方程，微分几何和数学物理。 例如，椭圆算子的泛函行列式是一个起源于谱理论和数学物理的问题，而我们考虑的特定问题的分析是一个导致四阶椭圆方程的变分问题。 相应的拉格朗日量是无界的，解的存在性及其定性性质是高度非平凡的。 类似的方程被用来模拟薄膜的性质。另一组问题的主要研究人员将探讨涉及某些完全非线性方程所产生的共形几何和理论的最佳运输。当底层流形为负弯曲时，这些方程的高阶正则性失效。虽然在某种意义上理解了这种缺乏规律性的结构原因（即，非线性如何导致某些估计的失败），没有奇点性质的几何描述。 几何与分析的相互作用至少可以追溯到世纪，至今仍是数学研究中一个重要而活跃的领域。经典的几何学是从我们对物理世界的某些性质的渴望中产生的，微分几何是为了理解弯曲空间的几何而发展起来的，例如，地球表面的曲率，或者广义相对论所预测的物质空间的曲率。在同样的方式，笛卡尔意识到，平面几何可以研究使用代数，所以微分几何可以研究使用技术分析，特别是微分方程。该项目的研究涉及几何和数学物理的不同问题，但其各个方面都通过数学分析在其研究中所发挥的作用而统一起来。

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.