Free Boundary Problems for Cell Motility and Other Applications

University Of Maryland, College Park, College Park MD

Investigators

Abstract

This project is devoted to the study of some mathematical problems that have important applications in physics and biology. One of the projects is concerned with the modeling of the motion of eukaryotic cells on a substrate. Cell motility is involved in key physiological processes such as wound healing and immunological response. One of the most remarkable characteristics of eukaryotic cells is their ability to reach and maintain an asymmetric shape in a seemingly spontaneous way, a phenomenon that leads to the sustained motion of the cell in a given direction (cell migration). While the biological processes involved are very complex, the principal investigator will develop and study some mathematical models (obtained as approximation of models proposed by biologists) that are both easier to analyze and faster to compute numerically. By identifying models that lead to cell migration, this work will help better understand what biological processes play a key role in cell motility. A different project is aimed at the study the fine properties of the solutions of optimal transportation problems. Optimal transportation problems are a class of mathematical problems that originated with the simple question of how to optimally allocate the production from a set of sources to a set of destinations in the cheapest (or most efficient) way. This field of mathematics has application in a variety of domains such as economics, data analysis, image processing etc. In addition to theoretical studies this research will contribute to the numerical computations of the solutions of these complex problems. Graduate students will be trained through active participation in the project. The first project described above involves free-boundary problems of Hele-Shaw type in which the usual smoothing/stabilizing effect of mean-curvature is balanced by the destabilizing effect of an active potential. The investigator will study symmetry breaking bifurcation phenomena characterized by the existence of nontrivial traveling wave solutions for such problems. Related free-boundary problems, arising in the modeling of congested crowd motion and in fluid dynamic will also be studied. The focus of the proposal is on models/regimes in which the forward and backward motions of the moving boundary occur via different mechanisms and at different time scales. In the field of optimal mass transportation, the focus is on the properties of optimal plans associated to measures that are discrete approximation of absolutely continuous measures. Such a framework is of great importance in many applications and in particular for numerical computations. A regularity theory for the associated Kantorovich potential will be developed. The final project is concerned with boundary conditions for nonlocal equations (e.g. fractional Laplace equation), which are notoriously more delicate than their local counterparts. The main goal of this project is derivation of new nonlocal Neumann boundary conditions, which are the macroscopic counterparts of classical microscopic boundary conditions in the kinetic theory of gas dynamic. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.