The research of the principal investigator will be focused on different mathematical areas. First of all, he plans to work on optimal transport theory. This problem, which consists in finding the cheapest way to transport a distribution of mass from one place to another, has recently found many applications in meteorology, biology and populations dynamics, or for instance to study the antenna design problem. The investigator has already worked on this subject for several years. He also intends to study Euler equations for incompressible fluids and to work on problems of semiclassical limit coming from quantum physics. Finally, he wishes to apply some of his recent results on some "improved" version of the classical isoperimetric inequalities to study the stability of the shapes of crystals under the action of an external potential.
The research described above has applications in many different areas: the optimal transport problem has obvious application to economics, but it has also shown to be very interdisciplinary, with links with other areas of mathematics like geometry, probability and partial differential equations, and also with physics and biology. Euler equations and semiclassical limits are classical problem in physics and quantum physics, and their mathematical study may increase deeper understanding of the physical phenomena themselves. Finally, the study of the rigidity of crystals under exterior potential should help to explain many phenomena which are currently observed in experiments but not yet completely understood.
In the course of the last three years, thanks to the support coming from this NSF grant, the PI has been able to devote a considerable amount of time working on the following problems: - Optimal transport theory: What is the cheapest way to transport a distribution of mass from one place to another? This problem, which has clearly important applications in economics and industry, has also been found to be useful to other areas of mathematics. First of all, many equations coming from methereology (like the semi-geostrophic equations), biology and population dynamics can be successfully studied using optimal transport theory. Moreover the optimal transport problem is also related to Monge-Ampère type equations, and this link can be exploited for instance to study the antenna design problem. The antenna design problem is the following: given a point source, build an antenna prescribing the intensity of the reflection along any given ray in a subdomain of the unit sphere. With the use of optimal transport theory, several important new results on the geometric properties of antenna reflectors have been recently obtained. - Incompressible Euler equations: The incompressible Euler equations describe the evolution of the velocity field of an incompressible fluid. Recently, many variational models have been introduced to study solutions of Euler equations and their hydrodynamic limit. Because of the applications of Euler equations to physics and engineering, understanding both qualitative and fine properties of solutions to these equations is a fundamental challenge both in mathematics and science. - Anisotropic surface energies and stability: The anisotropic perimeter of a set is a generalization of the classical notion of perimeter: here a different weight is given to every point on the boundary of the set, depending on the direction of its normal. Apart from its intrinsic geometric interest, the anisotropic perimeter arises as a model for surface tension in the study of equilibrium configurations of solid crystals with sufficiently small grains and constitutes the basic model for surface energies in phase transitions. Since perfect crystals are known to minimize such surface tensions, it is important to understand the stability of minimizers in order to be able to quantify how far the shape of a crystal is from the"perfect one" when some other forces (like the gravitational one) act on it. - Semiclassical limit: A natural question in quantum physics is to examine whether the evolution governed by the Schrodinger equationconverges to the classical Newtonian dynamics when one lets the Planck constant tend to zero. Mathematically, this problem presents serious difficulties. For instance, if one considers the Born-Oppenheimer quantum molecular dynamics then the interaction potential between particules is not smooth, so it is diffucult to rigorously justify the passage to the limit. The outcomes of this research have been impressive. Apart from several results which considerably improved the understanding of the phenomena that the models above attempt to reflect, among the most remarkable achievement of the PI that we would like to mention is a regularity result for the Monge-Ampère equation. This result, apart from its own mathematical interest, is the key tool to prove existence of distributional solutions to the semigeostrophic equations. The latter are a simple model used in meteorology to describe large scale atmospheric flows and they can be derived from the 3-d incompressible Euler equations, with Boussinesq and hydrostatic approximations, subject to a strong Coriolis force. Because of lack of regularity results for the Monge-Ampère equation, no global existence results for the "real" semigeostrophic system was known. Exploiting the above-mentioned regularity result for Monge-Ampère, the PI and his collaborators could provide a global in time existence of distributional solutions to the semigeostrophic equations in 2 and 3 dimensions, under very mild assumptions on the initial data. In addition to the scientific impact of the research, the proposed research activity has been an important and integral part of the PI's training program of undergraduate, graduate, and postdoctoral students. Many of the PI's PhD students and postdocs have been engaded in research in these areas and the results obtained have been widely disseminated via the publication of research papers and lecture notes, as well as through the developement of courses and seminars.
