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

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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.