Lower semicontinuity for functionals with (p, q)-growth in Carnot groups

摘要

Let Ω be a bounded open subset of ${mathbb{G}}$ , where ${mathbb{G}}$ is a Carnot group, and let ${u: Omega rightarrow mathbb{R}^d}$ be a vector valued function. We prove a lower semicontinuity result in the weak topology of the horizontal Sobolev space ${W^{1,p}_X(Omega,mathbb{R}^d)}$ , with p 1, of the integral functional of the calculus of variations of the type $$F(u)=intlimits_Omega f(Xu),dx$$ where f is a X-quasiconvex function satisfying a non-standard growth conditions and Xu is the horizontal gradient of u.