Many physical problems can be modelized using variational principles, such as energy or entropy which obeys a minimization , maximization or saddle-point law.
Starting from some classical examples such as the brachistochrone, the catenary, we will derive the mathematical model and provide mathematical tools for the existence of minimisers. Necessary and sufficient conditions will be discussed. In particular, we will prove the Tonelli Theorem.
For multiple integrals we will introduce Sobolev spaces and the corresponding notions that ensure lower semicontinuity. An insight into the regularity of minimizers will be discussed.

Classical examples
Introduction and definitions of some function spaces
Classical methods- Euler-Lagrange equations
Convex Analysis
Tonelli Theorem
Sobolev spaces
Direct methods in the Sobolev setting
Dirichlet integral and regularity
Nirenberg method
De Giorgi Theorem
Vectorial case
Semicontinuity and compactness theorems
Some comments about regularity