Theorems on lower semicontinuity and relaxation for integrands with fast growth

We prove theorems on the lower semicontinuity and integral representations of the lower semicontinuous envelopes of integral functionals with integrands $L$ of fast growth: $c_1G(|Du|)+c_2\leqslant L\leqslant c_3G(|Du|)+c_4$ with $c_3\geqslant c_1>0$ and $G\colon{[0,\infty[}\to{[0,\infty[}$ is an increasing convex function such that $vG'(v)/G(v)\to\infty$ as $v\to\infty$ and is increasing for large $v$. Repeating the results for the case of the standard growth $G(\cdot)={|\cdot|^p}$) the quasiconvexity of integrands characterizes the lower semicontinuity of integral functionals and their quasiconvexifications yield the integral functionals that are lower semicontinuous envelopes.