Isoperimetric Variational Techniques and Applications

Abstract

This project is at the interface between Nonlinear Functional Analysis, Con- vex Analysis and Di erential Equations. It concerns one of the most powerful methods often used to solve optimization problems with constraints; namely the Variational Method involving Isoperimetric conditions. As applications the exis- tence of in netely many periodic solutions of some 2nd order dynamical systems will be proven in the line of M.S. Berger[1]. Variational methods refer to proofs established by showing that a suitable auxilliary function attains a minimum or has a critical point (cf. De nition ...). In the former case, this can be viewed as a mathematical form of the princi- ple of least action in Physics and justi es why so many results in Mathematics are somehow related to variational techniques as they have their origin in the physical sciences. Their applications cover numerous theoretic as well as applied areas including optimization, Banach space geometry, nonsmooth analysis, eco- nomics, control theory and Game theory. But we shall focus on a branch linking minimization and periodic di erential equations. My interest in this subject has been steadily fascinated by the successive lectures delivered at the African University of Sciences and Technology by Prof. C. Chidume (Functional Analysis)[2], Dr. N. Djitte (Sobolev spaces and lin- ear elliptic partial di erential equations)[3], Dr G. Degla (Topics in Di erential Analysis)[4] and Prof. Thibault (Convex Analysis)[5].