RUI: PDE and Geometry in non-smooth spaces

This award supports a project which investigates topics in the theory of partial differential equations in the setting of non-smooth spaces. Partial differential equations provide a powerful mathematical tool to gain insights about equilibrium states of complex physical systems which arise as solutions of certain equations. The properties of the solutions to these equations depend on a “background geometry” that models physical features such as the non-homogeneity of materials or the presence of constraints (such as the constraints inherent in the motion of a robotic arm). In many important physical applications, one encounters non-smooth geometries (for instance, fractals) which differ fundamentally from the familiar geometry of Euclidean space, so that standard notions from calculus must be reformulated from a broader perspective. One of the most ubiquitous instances of such “background geometry” is known as sub-Riemannian geometry, which models spaces in which motion is possible only along a given set of directions. This non-smooth geometry is widely useful in modeling physical phenomena, for example, in robotics, quantum mechanics, and neuroscience. This project will also provide opportunities for undergraduate and graduate students to work on research projects arising from the proposed work. The PI will study sub-Riemannian analogues of the curve shortening flow; the regularity of solutions of certain degenerate elliptic parabolic PDE and non-local PDE in the general setting of certain metric spaces endowed with a doubling measure. The common thread between these investigations is the interplay between the non-smooth structure of the space and the behavior of solutions of equations describing critical points of interesting energy functionals. Some of the proposed research will provide a theoretical basis for implementing numerical simulations of real-world systems. 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.