Weak Lower Semicontinuity of Integral Functionals and Applications

摘要

Minimization is a recurring theme in many mathematical disciplines ranging from pure to applied. Of particular importance is the minimization of integral functionals, which is studied within the calculus of variations. Proofs of the existence of minimizers usually rely on a fine property of the 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 established by C. B. Morrey, Jr. [Pacific J. Math., 2 (1952), pp. 25{53] in 1952 and N. G. Meyers [Trans. Amer. Math. Soc., 119 (1965), pp. 125{149] in 1965. We recapitulate the development of this topic from these papers onwards. 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. In addition, we emphasize some recent progress in lower semicontinuity of functionals along sequences satisfying differential and algebraic constraints that can be used in elasticity to ensure injectivity and orientation-preservation of deformations. Finally, we outline generalizations of these results to more general first-order partial differential operators and make some suggestions for further reading.