Some historical facts and a few classical examples: the Fermat principle, the Newton problem, the brachistochrone, the minimal surface of revolution an minimal surfaces; the Dirichlet integral and its variants. 

A brief introduction to the fundamental problem of minimisation of functional in integral form and the deduction of the Euler-Lagrange necessary conditions. Extension to the case of minimisation the Dirichlet integral in a convex set of functions and the variational inequality for the obstacle problem.

Next lecture: Friday, 28 February at 14:00 (room to be confirmed)
Exercise: 1 - Read the Preliminaries chapter and the §§ 2.1, 2..2 and 2.3 of Dacorogna Book;Exercise: 2 - Write the variational inequality that is a necessary condition for a solution of a minimisation problem of the 
general functional   (15), page 460,  of Chap. 8 of Evans PDEs book 
in a convex set where the vector minimiser  u exists .