NSF Award
2042654
Elliptic differential equations are used to describe a wide variety of physical phenomena. A main goal of this project is to study a particular class of these equations that can be used to model the behavior of vibrations of a drum or an inhomogeneous material, the temperature distribution of a thermal conductor, or the distribution of the random motion of particles moving in a region of fluid. Other than in a few simple model cases, this behavior is still far from understood. By reformulating these problems in terms of partial differential equations, this project will study this behavior. One particular question of interest is how does the shape of a drum influence which part of the drum its vibrations are typically localized to. Another aim is to study free boundary problems. A free boundary is the region separating two different materials, such as the interface between water and the air in the ocean or between an insulating material and the air. The free boundary can be modeled via solving differential equations, and understanding its shape and the scale at which it appears smooth has applications to optimal shape design, electromagnetism, and fluid flow. This project contributes to the development of the US workforce through mentoring of undergraduate students.<br/><br/><br/>In this project, the PI will study eigenvalues and eigenfunctions on convex domains in Euclidean space and the sphere. A main goal of the project is to further the understanding of the level sets of the eigenfunction. The starting point is a quantitative property, namely the convexity of the level sets, and the aim is to use and develop techniques from elliptic and differential geometry theory to establish quantitative properties involving their shape and location with the domain. This would lead to understanding the region of the domain where the eigenfunctions localize in cases where no explicit formulae are available. Another part of the project involves variants of the classical Friedland-Hayman inequality concerning eigenvalues on the sphere. This will involve developing tools from convex geometry, isoperimetric inequality theory, and Brenier's optimal transportation mappings. This will be applied to prove regularity properties of minimizers of associated free boundary problems.<br/><br/>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.
