The central problem of the Calculus of Variations consists in finding a minimizer to given functional F, the latter being often of integral type.  

The birth of the Calculus of Variations dates back to the seventeenth century. Indeed minimization problems for integral functionals are related to classical problems in mechanics, such as the Brachistochrone problem (Bernoulli 1697), as well as to problems in geometrical optics, such as the Fermat principle (Fermat 1662).

Despite its old history, in the last decades there has been a renewed an ever increasing interest in the Calculus of Variations which was mainly originated by the study of variational models in nonlinear elasticity and, more in general, in continuum mechanics.

The main object of this course will be to study the existence of solutions to minimization problems for integral functionals  over a ``suitable'' functional space. We will focus on two different approaches: the classical  and the direct method. More specifically, as in the finite-dimensional case, one way of studying a minimization problem consists is finding the zeroes of the ``derivative'' of  F, hence to solve F'(u)=0 known as the Euler-Lagrange equation, and then studying the positivity of the second variation around the solutions. To do so there are various necessary or sufficient conditions, namely Weierstrass, Legendre or Jacobi conditions. This is the idea behind the so-called classical method.

The direct method deals instead directly with the functional F; here the main idea is to find conditions on F (and hence on its integrand) that make it possible to prove the existence of minimizers via a suitable extension of the Weierstrass theorem to the infinite-dimensional setting.

Um die Vorlesung als Bestanden eintragen zu lassen, benötigen Sie die regelmäßige Teilnahme an der Vorlesung und die Übungen müssen ebenfalls bestanden sein.

Kurs im HIS-LSF