Calculus of Variations

Variational methods have a long tradition in the history of Mathematics. The Greeks studied isoperimetric problems to find "optimal forms". In 1662 Fermat postulated that light will always follow the path of minimal time, and deduced from this principle the refraction laws. In 1744 Euler even postulated that "every effect in nature follows a maximum or minimum principle". This affirmation may be exaggerated, but it is certainly true that the Calculus of Variations provide a powerful instrument to study many problems in mathematics, physics and the applied sciences. Among the most important applications are: existence of geodesics, surfaces of minimal area, periodic solutions of N-body problems, existence of solutions for nonlinear elliptic PDE. The course provides an introduction to the modern theory of Calculus of Variations.

Acquisition of the basic notions and techniques in the theory of Calculus of Variations: minimization, deformations, problems of compactness, relations between topology and critical points. Study of the relations between critical point theory and partial differential equations.