Variational convergences for functionals and differential operators depending on vector fields

Venue: The event will take place online through the ZOOM platform. To get the access codes please contact the secretary office (phd.maths [at] unitn.it)
Time: 3.30 p.m.

Alberto Maione - PhD in Mathematics, University of Trent

Abstract:
In this seminar, we present results concerning variational convergences for functionals and differential operators depending on a family of locally Lipschitz continuous vector fields X. This setting was introduced by Folland and Stein and has recently found numerous applications in the literature. The convergences taken into account date back to the 70’s and are Γ-convergence, introduced by Ennio De Giorgi and Tullio Franzoni, dealing with functions and functionals, and H-convergence, whose theory was initiated by François Murat and Luc Tartar and which deals with differential operators. 
The main result presented today, under a linear independence condition on the family of vector fields X, is a Γ-compactness theorem and ensures that sequences of integral functionals depending on vector fields, with standard regularity and growth conditions, Γ-converge in the strong topology of Lp, up to subsequences, to a functional belonging to the same class. 
As an interesting application of the Γ-compactness theorem, we show that the class of linear differential operators in X-divergence form is closed in the topology of the H-convergence. The variational technique adopted to this aim relies on a new approach recently introduced by Nadia Ansini, Gianni Dal Maso and Caterina Ida Zeppieri.

Supervisors: Francesco Serra Cassano, Andrea Pinamonti