Adventures with the mountain pass theorem: existence, multiplicity, uniqueness

Some of the most powerful tools for the study of differential equations come from the Calculus of Variations, beginning with Fermat's Least-action Principle and Bernoulli's solution to the Brachistochrone problem through Dirichlet's Principle and minimal surfaces. The celebrated Mountain-Pass Theorem of Ambrosetti and Rabinowitz is a simple but elegant technique for finding non-minimizing critical points, which has had enormous applicability in ordinary and partial differential equations. While it is usually presented as a tool for proving existence of solutions, I will give some examples of its versatility in proving multiplicity, the qualitative form of solutions, and even uniqueness!

Refreshments will be served in Math Annex Room 1115, 3:15 p.m.