Minimization is a reoccurring theme in many mathematical disciplines rangingfrom pure to applied ones. Of particular importance is the minimization ofintegral functionals that is studied within the calculus of variations. Proofsof the existence of minimizers usually rely on a fine property of the involvedfunctional called weak lower semicontinuity. While early studies of lowersemicontinuity go back to the beginning of the 20th century the milestones ofthe 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 attentionis paid to signed integrands and to applications in continuum mechanics ofsolids. In particular, we review the concept of polyconvexity and specialproperties of (sub)determinants with respect to weak lower semicontinuity.Besides, we emphasize some recent progress in lower semicontinuity offunctionals along sequences satisfying differential and algebraic constraintswhich have applications in elasticity to ensure injectivity andorientation-preservation of deformations. Finally, we outline generalization ofthese results to more general first-order partial differential operators andmake some suggestions for further reading.
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