The Calculus of Variations researches optimal states for functionals with real values. There are many disparate problems falling within this broad theme. For instance, we may mention the energy minimisation in a mechanical system in which equilibrium configurations are sought, the minimisation of production costs, the optimisation of a structure's performance. Minimisation problems naturally appear within Euclidean geometry, too (isoperimetric problems) and in differential geometry, and in all fields of the "hard" sciences, as well as in the economic and social ones.

Starting with Hilbert and Tonelli and up until today, the techniques became more and more refined. By developing the concepts of semi-continuity and coercivity, the decisive contribution of Ennio De Giorgi and of his school was attained. Most of the research activity and of the mathematical techniques implemented by the group indeed derive their origin and foundation from De Giorgi's work.

In particular, the group deals with the identification and characterisation of variational models for studying thin mono-dimensional and bi-dimensional material structures, including non-homogeneous or anisotropic materials, as limitations of three-dimensional structures when one or two dimensions tend to zero (dimensionality reduction techniques).

Other research themes concern stationary and evolutionary problems regarding variational models for the development of fractures in homogeneous materials. We plan to deduce, starting from general physical principles, a rigorous mathematical formulation of a dynamical evolutionary model for the development of fractures in elastic solids, which will also take into account the kinetic energy caused by vibrations.

Moreover, the group is interested in variational models governed by non-local differential operators (fractional Laplacians), more suitable for modelling phenomena featuring random stochastic motions (random walks) or long-range interactions. The understanding of the dependence of such operators with regard to domain variations is still incomplete. The group's research activity includes issues related with Gamma-convergence results for such operators, for which only a few partial results are available.