VIGRE Minicourse on Variational Methods and Nonlinear PDE

Department of Mathematics, University of Utah

May 28 - June 8, 2002





Working groups

Working Groups

We have divided students into the following four working groups. Each group will give a presentation during the second week of the minicourse, as indicated on the schedule.

  1. Bernoulli's solution of the brachistochrone problem was an early triumph in the calculus of variations, and others (including Euler) subsequently succeeded in solving isoperimetric problems using similar indirect methods. This group will study these indirect methods and their application to the brachistochrone and isoperimetric problems.

    Group: Le, Simon

    Project: PostScript || Adobe PDF

  2. A dominant technique in early variational approaches was Dirichlet's principle, which lost favor after some mathematicians (notably Weierstrass) pointed out its weaknesses. This group will discuss Dirichlet's principle, its flaws, and its salvation via direct minimization methods (e.g., the role of lower semicontinuity and the identification of appropriate function spaces).

    Group: Greer, Koizumi, Messer

    Project: PostScript || Adobe PDF

  3. Many interesting problems in ordinary differential equations involve Sturm-Liouville operators, and the spectra of such operators have an elegant variational characterization. This characterization extends to higher-dimensional problems as well, where it often plays a crucial role. This group will discuss variational techniques for Sturm-Liouville problems.

    Group: Cormani, Rudd, Ryham

    Project: PostScript || Adobe PDF

  4. Dirichlet's principle concerns minimizers of functionals; it has proven extremely useful to study other critical points of functionals, such as saddle points. An early success in this direction was Ljusternik and Schnirelman's work on geodesics, which spawned the study of global analysis. This group will study geodesics, starting with the simple problem of geodesics on a sphere, and then discuss problems related to Ljusternik--Schnirelman theory.

    Group: Bisgard, Chu, Sim

    Project: PostScript || Adobe PDF