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.
The research project completed includes solving problems using harmonic analysis techniques with a view toward applications to nonlinear partial differential equations. One part of the project investigates the sharp geometric inequalities of Moser-Trudinger and Adams type, considers their applications to nonlinear partial differential equations with nonlinearity of exponential growth, and explores applications of these ideas in a variety of geometric settings (e.g., Euclidean space, Riemannian manifolds, Sub-Riemannian manifolds). There are circumstances where symmetrization properties do not hold and known methods do not work. Such problems are important in both analysis and geometry. The PI, jointly with his graduate students and collaborators, has succeeded in deriving the sharp constants and existence of extremal functions in a number of important cases and established the existence and multiplicity of nonnegative solutions. Nevertheless, there are still many challenging problems that remain open in this direction. Another part of the project concerns multi-parameter Hardy space and singular integral operator theory. The PI, together with his students and collabortaors, has developed a discrete multi-parameter Littlewood-Paley theory and applied it to multi-parameter Hardy space theory and boundedness of multi-parameter singular integral operators on such spaces. Our method and approach avoids deep Journe's covering lemma and applies to many important multi-parameter settings. Projcts finished and methods developed here will have important impact on further development in the direction of analysis and geometry. Findings and new tools discovered in this project will lead to some new development in the area of classical harmonic analysis, partial differential equations as well as other branches of mathematics. Harmonic analysis and Partial differential equations are important tools in applications of sciences and engineering. Discoveries in the projects carried out by the PI and his students and collaborators are useful to the understanding of phenomenon appearing in other disciplines. 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. The PI has made many presentations on findings in the sponsored project at national and international conferences. This help to disseminate the new development into a wide mathematical community. Moreover, this project has a substantial training and educational component. It finely integrates research together with education. Many graduate students (including two female students) actively participated in this project by receiving research training under the supervision of the principal investigator. Students were financially supported by this grant to attend many meetings of the American Mathematical Society and other national and international conferences to present their results and findings. Through such participation of conferences, students have the opportunities to exchange ideas and interact with other well established mathematicians. By carrying out the proposed project, students are able to grow into strong young mathematicians. The PI has completed and published in high quality journals about 25 joint research papers with his Ph.D students in the past four years. Ph.D. students participated actively in the projects to pursue their degrees. This helps the training of mathematicians of next generation, in particular women mathematicians. Students trained and supervised by the PI have succeeded very well in their career endeavor. Among his successful students, a female student who graduated in 2007 is now a tenured Associate Professor at University of Wisconsin at Milwaukee; one of his Ph.D students is a Risk Analyst at the Ford Motor Credit Company; another female student is an Assistant Professor at Nanjing Universty in China; his Ph.D student who graduated in 2012 is currently holding a three year postdoctoral position at the Australian National University at Canberra; his student who received his Ph.D. in 2013 became a J. J. Sylvester Assistant Professor at John Hopkins University; another female stduent who received her Ph.D in 2013 is now a tenure-track Assistant Professor at the Ball State University. The PI is currently supervising seven Ph.D students at Wayne State University, including a female student. He is also mentoring a postdoctor fellow at Wayne State University. They have been frequently sponored by this grant to attend many mathematical conferences and do presentations at these conferences. The PI has many collaborators on the project, both young and well-established. Some of the projects involve joint effort among mathematicians from other countries and their visits to Wayne State University have stimulated the mathematical research interest of graduate students in closely related research areas and enhance the graduate training progarm in mathematics.
