Geometric Variational Problems and Rearrangement Inequalities

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AbstractAward: DMS-0308040Principal Investigator: Almut BurchardTwo sets of problems are proposed, one related with elasticcurves, and the other with rearrangement inequalities. The firstset of problems concerns the dynamics of an elastic curve movingin three-dimensional space, under forces determined by itscurvature and subject to the constraint that it is inextensible.The main question is whether solutions starting from sufficientlysmooth initial data exist for all time, or whether they maybecome singular in finite time. For the long-time evolution ofan infinite curve, solitary waves are expected to play animportant role. The boundary value problem for the tension willbe studied in detail, with the goal of obtaining sharp bounds forthe tension in terms of the position and velocity of the curve.In the second set of problems, the PI proposes to settle aconjectured quantitative version of the classical rearrangementinequality for the Coulomb energy. Furthermore, the cases ofequality in the Brascamp-Lieb-Luttinger inequality for multipleintegrals will be investigated. In addition, the PI participatesin two interdisciplinary collaborations, on Statistical NetworkCalculus and its applications to data traffic (with colleaguesfrom Computer Science and Systems Engineering), and on theclustering of proteins on membranes within polarized cells (withcolleagues from Biology).The elastic curve equations described above provide a simpleexample of a class of nonlinear wave equations with geometricforces and geometric constraints. The analytic difficultiesencountered in this problem, and the techniques used to overcomethem, are expected to be relevant for more complex mechanicalsystems with geometric forces. While the work is geometricallymotivated, the dynamics of elastic wires are clearly relevant tostructural questions in Mechanical Engineering. The basicwell-posedness questions addressed here could have implicationsfor the convergence of numerical solution schemes. Rearrangementinequalities are among the few geometric tools for minimizingnonlinear, non-convex functionals. The proposed work strives tosharpen those tools and extend their applicability. Implicationsinclude the stability of symmetric galaxy configurations andother dynamical stability problems. A notable contribution ofthe applied collaborations is that they expose researchers inBiology and Engineering to the everyday use of simple, butrigorous Mathematics. The Network Calculus results center onquestions of admission control and service guarantees in largedata networks. The biological work has implication for themechanics of protein sorting, which is crucial to the function ofthe cell at a basic level.