Harmonic analysis and partial differential equations: sharp geometric inequalities, fully nonlinear equations and applications

调和分析和偏微分方程：尖锐的几何不等式、完全非线性方程和应用

• 批准号: 0901761
• 负责人: Guozhen Lu
• 金额: $ 23.97万
• 依托单位: Wayne State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-11-15 至 2013-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901761&HistoricalAwards=false
• 关键词: Harmonic; analysis; partial; differential; equations

This research project involves topics that use harmonic analysis techniques with a view toward applications to nonlinear partial differential equations. One part of the project investigates the sharp geometric inequalities of Sobolev, Moser-Trudinger, and Adams, considers the existence of extremal functions for them, and explores applications of these ideas in a variety of geometric settings (e.g., Euclidean space, Riemannian manifolds, the Heisenberg group, spheres in complex space, CR-manifolds). Such problems are important in both analysis and geometry. The principal investigator, jointly with his collaborators, has succeeded in deriving the sharp constants and existence of extremal functions in a number of important cases. Nevertheless, there are still many challenging problems that remain open. Another group of problems is concerned with the existence, uniqueness, and regularity of solutions to nonlinear partial differential equations, in particular, the inhomogeneous infinity Laplacian and degenerate Monge-Ampere equations.Harmonic analysis and nonlinear partial differential equations are central areas of modern mathematics. They have found applications in numerous disciplines, including engineering (such as vibration and noise reduction), stochastic control and optimization, game theory, physics, chemical combustion, mass transport, human vision and other topics in the life and medical sciences. Graduate students will participate in this project by receiving research training under the supervision of the principal investigator.

这项研究项目涉及使用调和分析技术的主题，以期应用于非线性偏微分方程。项目的一部分研究了Sobolev，Moser-Trudinger和Adams的尖锐几何不等式，考虑了它们的极值函数的存在性，并探索了这些思想在各种几何环境中的应用(例如，欧几里德空间，黎曼流形，Heisenberg群，复空间中的球面，CR-流形)。这类问题在分析和几何中都很重要。这位首席研究员与他的合作者一起，在一些重要的案例中成功地推导出了尖端常数和极值函数的存在性。然而，仍然有许多具有挑战性的问题仍然悬而未决。另一类问题是关于非线性偏微分方程解的存在唯一性和正则性，特别是非齐次无穷拉普拉斯方程和退化的Monge-Ampere方程。调和分析和非线性偏微分方程组是现代数学的中心领域。它们在许多学科中都有应用，包括工程学(如减振和降噪)、随机控制和优化、博弈论、物理学、化学燃烧、质量传输、人类视觉以及生命和医学科学中的其他主题。研究生将通过在首席研究员的指导下接受研究培训来参与这个项目。