  Minimization is a reoccurring theme in many mathematical disciplines ranging
from pure to applied ones. Of particular importance is the minimization of
integral functionals that is studied within the calculus of variations. Proofs
of the existence of minimizers usually rely on a fine property of the involved
functional called weak lower semicontinuity. While early studies of lower
semicontinuity go back to the beginning of the 20th century the milestones of
the modern theory were set by C.B. Morrey Jr. in 1952 and N.G. Meyers in 1965.
We recapitulate the development on this topic from then on. Special attention
is paid to signed integrands and to applications in continuum mechanics of
solids. In particular, we review the concept of polyconvexity and special
properties of (sub)determinants with respect to weak lower semicontinuity.
Besides, we emphasize some recent progress in lower semicontinuity of
functionals along sequences satisfying differential and algebraic constraints
which have applications in elasticity to ensure injectivity and
orientation-preservation of deformations. Finally, we outline generalization of
these results to more general first-order partial differential operators and
make some suggestions for further reading.
