Weak minima of integral functionals in Carnot-Carathéodory spaces (Q2467926)

The paper deals with the regularity for weak minimizers of integral functional of the form \[ F(u)= \int_{\Omega} F(Xu)\,dx \] with \(u\) defined in \(\Omega \subset R^n\), \(F=F(\xi)\) a continuous function satisfying \(p\)-growth condition and \(X=(X_1,\dots,X_k)\) is a family of vector fields, satisfying the so-called Hörmander conditions. More precisely, if \(u \in W^{1,r}_{X,\text{loc}}(\Omega)\), with \(r_1 \leq r <p\), is a weak minimizer of \(F\), then \(u \in W^{1,p}_{X,\text{loc}}(\Omega)\). The paper generalizes some regularity results obtained in the classical situation when \(Xu=Du\), by using a suitable change of metric in \(R^n\) and the theory of Carnot-Carathéodory space.