X-quasiconvexity in Carnot Groups and lower semicontinuity results

We characterize lower semicontinuity of integral functionals of the calculus of variations in the setting of Carnot Groups. Accordingly, we introduce the notion of X-quasiconvexity, that is referred to the family of H\"ormander vector fields, associated to the Group. The relevance of this general framework lies on the fact that many physical phenomena induce in a natural way an associated sub-Riemannian structure. For example, one can think of Berry's phase problem, a swimming micro-organism, the perceptual completion in the visual cortex . We consider also the problem of relaxation. We are able to give a representation formula for the X-quasiconvex envelope of the functional .