Lower semicontinuity of integrals of the calculus of variations in Cheeger–Sobolev spaces

A necessary condition called $$H_\mu ^{1,p}$$-quasiconvexity on p-coercive integrands is introduced for the lower semicontinuity with respect to the strong convergence of $$L^p_\mu (X;\mathbb {R}^m)$$ of integral functionals defined on Cheeger–Sobolev spaces. Under polynomial growth conditions it turns out that this condition is necessary and sufficient.