In this talk, I discuss about variational convergences for functionals and differential operators depending on a family of Lipschitz continuous vector fields X. This family is chosen in a way that the associated anisotropic Sobolev spaces, in the sense of Folland and Stein, satisfy a global Poincaré inequality and a Rellich-Kondrachov-type theorem; examples of such families are the euclidean gradient, the Grushin gradient and the Heisenberg horizontal gradient. However, our setting does not only consider vector fields satisfying the Hörmander condition.
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 G-convergence, whose theory was initiated by Ennio De Giorgi and Sergio Spagnolo, for the case of symmetric matrices, and later known as H-convergence, after the extension to the non-symmetric and nonlinear cases by François Murat and Luc Tartar. This kind of convergence deals with differential operators.
The main result presented today is a Γ-compactness theorem. It ensures that sequences of integral functionals depending on the family X, with standard regularity and growth conditions, Γ-converge in the strong topology of Lᵖ (p > 1) to a functional belonging to the same class and, as an interesting application, 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 last aim relies on a new approach recently introduced by Nadia Ansini, Gianni Dal Maso and Caterina Zeppieri.
These results are obtained in collaboration with Andrea Pinamonti and Francesco Serra Cassano (University of Trento).