Variational Problems and Dynamics

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The proposed research program is focused on the interplay of dynamics and variational problems, with a special emphasis on quantitative stability estimates for sharp geometric inequalities. Such estimates and inequalities provide an effective means to derive properties of solutions of evolution equations and likewise, evolution equations can be used to derive variational inequalities. Exploiting this interplay has been very fruitful in the past work of the investigator, who plans to approach various problems in this framework. One is to prove new sharp non-local version of the Gagliardo-Nirenberg-Sobolev inequalities that arise in connection with the fast diffusion equation and porous medium equation. Such equations arise in the modeling of many phenomena, physical to biological, and the analytic problems to be investigated are chosen for their potential broader impact as well their intrinsic analytic interest. Another such problem concerns quantitiative stability for the Brun-Minkowski inequality and its application in the study of phase transitions. A similar philosophy applies as well to certain problems in kinetic theory, where the plan is to derive quantitative estimates on speed of approach to equilibrium for some inhomogeneous master equations of Kac type that have recently been used to model not only physical phenomena, but also phenomena in population biology.Many phenomena in science and technology can be modeled by evolution equations. One example that is the object of research in this proposal is the Keller Segel system, which models the aggregation, or the absence thereof, in the motion of bacteria. Understanding the behavior of solutions of these equations is both biologically and mathematically interesting. In particular, it is often observed that systems of many interacting agents (such as bacteria) or particles, either classical or quantum mechanical, evolve toward an equilibrium, and they do this at a certain speed. Determining the the speed of this process is vital to the understanding of many such models. This proposal concerns research on obtaining themathematical keys to such problems and their application in a broad range of fields. A central focus is solving certain maximization problems, and proving theorems asserting that any input that produces nearly maximal output must be close, in a an appropriate sense, to input that gives the exact maximum. As recent research of the investigator has shown, such theorems provide an effective means to unlock the information in models of the type described above, and the research proposed here will further this progress.