Principal Investigator: Almut Burchard
Two sets of problems are proposed, one related with elastic curves, and the other with rearrangement inequalities. The first set of problems concerns the dynamics of an elastic curve moving in three-dimensional space, under forces determined by its curvature and subject to the constraint that it is inextensible. The main question is whether solutions starting from sufficiently smooth initial data exist for all time, or whether they may become singular in finite time. For the long-time evolution of an infinite curve, solitary waves are expected to play an important role. The boundary value problem for the tension will be studied in detail, with the goal of obtaining sharp bounds for the tension in terms of the position and velocity of the curve. In the second set of problems, the PI proposes to settle a conjectured quantitative version of the classical rearrangement inequality for the Coulomb energy. Furthermore, the cases of equality in the Brascamp-Lieb-Luttinger inequality for multiple integrals will be investigated. In addition, the PI participates in two interdisciplinary collaborations, on Statistical Network Calculus and its applications to data traffic (with colleagues from Computer Science and Systems Engineering), and on the clustering of proteins on membranes within polarized cells (with colleagues from Biology).
The elastic curve equations described above provide a simple example of a class of nonlinear wave equations with geometric forces and geometric constraints. The analytic difficulties encountered in this problem, and the techniques used to overcome them, are expected to be relevant for more complex mechanical systems with geometric forces. While the work is geometrically motivated, the dynamics of elastic wires are clearly relevant to structural questions in Mechanical Engineering. The basic well-posedness questions addressed here could have implications for the convergence of numerical solution schemes. Rearrangement inequalities are among the few geometric tools for minimizing nonlinear, non-convex functionals. The proposed work strives to sharpen those tools and extend their applicability. Implications include the stability of symmetric galaxy configurations and other dynamical stability problems. A notable contribution of the applied collaborations is that they expose researchers in Biology and Engineering to the everyday use of simple, but rigorous Mathematics. The Network Calculus results center on questions of admission control and service guarantees in large data networks. The biological work has implication for the mechanics of protein sorting, which is crucial to the function of the cell at a basic level.
