Lower semicontinuity and relaxation for integral functionals with $p(x)$- and $p(x,u)$-growth

We consider the questions of lower semicontinuity and relaxation for the integral functionals satisfying the $p(x)$- and $p(x,u)$-growth conditions. Presently these functionals are actively studied in the theory of elliptic and parabolic problems and in the framework of the calculus of variations. The theory we present rests on the following results: the remarkable result of Kristensen on the characterization of homogeneous $p$-gradient Young measures by their summability; the earlier result of Zhang on approximating gradient Young measures with compact support; the result of Zhikov on the density in energy of regular functions for integrands with $p(x)$-growth; on the author's approach to Young measures as measurable functions with values in a metric space whose metric has integral representation.