1 - Variational principles

Summary

A man is like a fraction whose numerator is what he is and whose denominator is what he thinks of himself. The larger the denominator the smaller the fraction.

Leo Tolstoy (1828–1910)

Variational principles are very powerful techniques that exist at the interface between nonlinear analysis, calculus of variations, and mathematical physics. They have been inspired by and have deep applications in modern research fields such as geometrical analysis, constructive quantum field theory, gauge theory, superconductivity, etc.

In this chapter we briefly recall the main variational principles which will be used in the rest of the book, such as Ekeland and Borwein–Preiss variational principles, minimax- and minimization-type principles (the mountain pass theorem, Ricceri-type multiplicity theorems, the Brezis–Nirenberg minimization technique), the principle of symmetric criticality for nonsmooth Szulkin-type functionals, as well as Pohozaev's fibering method.

Minimization techniques and Ekeland's variational principle

Many phenomena arising in applications such as geodesics or minimal surfaces can be understood in terms of the minimization of an energy functional over an appropriate class of objects. For the problems of mathematical physics, phase transitions, elastic instability, and diffraction of light are among the phenomena that can be studied from this point of view.