Geometric Inequalities and Fully Nonlinear Elliptic Equations

几何不等式和完全非线性椭圆方程

• 批准号: 1547878
• 负责人: Yi Wang
• 金额: $ 1.12万
• 依托单位: Johns Hopkins University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-09-01 至 2015-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1547878&HistoricalAwards=false
• 关键词: Geometric; Inequalities; Fully; Nonlinear; Elliptic

The Principal Investigator (PI) will study various geometric inequalities and their corresponding nonlinear partial differential equations in the context of conformal geometry and the geometry of submanifolds. One theme plans to investigate the effect of the higher order curvatures on the validity of the isoperimetric inequality. In particular, she intends to quantitatively analyze the interaction between the Q-curvature and the isoperimetric constant. The PI also proposes to study curvature inequalities of different orders for embedded submanifolds. These inequalities, originally considered in the context of convex geometry, have recently known to be valid on a very large class of non-convex domains. The PI's investigation strives to look for the full generality of such inequalities. In the meanwhile, she also aims to develop new skills to understand the corresponding fully nonlinear elliptic partial differential equations that arise naturally in the problem.In the proposed study, the PI's research interest lies at the intersection of conformal geometry, the geometry of submanifolds and partial differential equations (PDEs). The study of geometric inequalities and geometric PDEs focuses on conformal invariants, which form an important machinery from physicists' point of view and have found applications to fundamental principles in mathematical physics. The research project to generalize classical results of convex geometry will improve our understanding on the rigidity of the established theory and will shed light on a greatly larger scope of its application.

主要研究者（PI）将研究各种几何不平等现象及其相应的非线性偏微分方程，并在共形几何形状和亚策略的几何形状中。一个主题计划研究高阶曲率对等级不平等的有效性的影响。特别是，她打算定量分析Q狂热与等值恒定之间的相互作用。 PI还建议研究嵌入的亚曼叶夫的不同顺序的曲率不平等。这些不等式最初在凸几何形状的背景下考虑，最近已知在一类非常大的非convex域上有效。 PI的调查致力于寻找这种不平等的一般性。与此同时，她还旨在发展新技能，以了解问题自然出现的相应非线性椭圆形偏微分方程。在拟议的研究中，PI的研究兴趣在于保形几何形状的交集，Submanifolds的几何形状，Submanifolds的几何形状和偏微分方程（PDES）。几何不平等和几何PDE的研究集中在形成物理学家的角度的重要机制上，并已将应用于数学物理学的基本原理。概括凸几何结果的经典结果的研究项目将提高我们对既定理论的刚性的理解，并将阐明其应用的更大范围。