Analytical and geometrical problems in calculus of variations and partial differential equations

University Of Texas At Austin, Austin TX

Investigators

Abstract

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.