Advanced Topics in Calculus of Variations

The aim of this course is to provide a modern and rigorous overview of the analytical techniques used to study elliptic partial differential equations through variational methods. At the core of this theory lies the possibility of interpreting weak solutions of elliptic equations with variational structure as critical points of suitable functionals defined on Sobolev spaces.
The course will explore in depth the techniques of the direct method in the calculus of variations, addressing constrained minimization problems and introducing the notion of natural constraints.
It will then move on to the analysis of several minimax methods, which allow the identification of nontrivial critical points of functionals that are not necessarily minima. Classical results such as the Mountain Pass Theorem and the Linking Theorems will be presented and applied to the study of nonlinear elliptic equations, both on bounded and unbounded domains, with particular attention to problems involving critical growth and phenomena related to loss of compactness.

Knowledge of the general principles of the calculus of variations and critical point theory, in particular the constrained minimization and the minimax principle. Ability to rigorously formulate a typical minimization problem (functional setting, energy functional) and to identify its main features and difficulties. Ability to apply the general theory to specific problems. Knowledge of some techniques and tools from the theory of partial differential equations.