Variational problems for sub–Finsler metrics in Carnot groups and Integral Functionals depending on vector fields

The first aim of this PhD Thesis is devoted to the study of geodesic distances defined on a subdomain of a Carnot group, which are bounded both from above and from below by fixed multiples of the Carnot–Carathéodory distance. Then one shows that the uniform convergence, on compact sets, of these distances can be equivalently characterized in terms of Gamma-convergence of several kinds of variational problems. Moreover, it investigates the relation between the class of intrinsic distances, their metric derivatives and the sub-Finsler convex metrics defined on the horizontal bundle. The second purpose is to obtain the integral representation of some classes of local functionals, depending on a family of vector fields, that satisfy a weak structure assumption. These functionals are defined on degenerate Sobolev spaces and they are assumed to be not translations-invariant. Then one proves some Gamma-compactness results with respect to both the strong topology of L^p and the strong topology of degenerate Sobolev spaces.