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

本研究课题涉及调和分析技术在非线性偏微分方程中的应用。该项目的一部分研究了Sobolev，Moser-Trudinger和亚当斯的尖锐几何不等式，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设置中的应用（例如，欧几里得空间，黎曼流形，海森堡群，复空间中的球面，CR-流形）。这类问题在分析和几何学中都很重要。首席研究员，与他的合作者，已成功地推导出尖锐的常数和存在的极值函数在一些重要的情况下。然而，仍然有许多挑战性的问题尚未解决。另一类问题涉及非线性偏微分方程，特别是非齐次无穷拉普拉斯方程和退化Monge-Ampere方程解的存在性、唯一性和正则性。调和分析和非线性偏微分方程是现代数学的中心领域。他们已经在许多学科中找到了应用，包括工程（如振动和降噪），随机控制和优化，博弈论，物理学，化学燃烧，质量运输，人类视觉和生命和医学科学中的其他主题。研究生将通过在主要研究者的监督下接受研究培训来参与本项目。

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.