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Journal Article

Weak Lower Semicontinuity of Integral Functionals and Applications

Benešová B., Kružík Martin

: SIAM Review vol.59, 4 (2017), p. 703-766

: GA14-15264S, GA ČR, GF16-34894L, GA ČR, DAAD-16-14, GA AV ČR

: calculus of variations, weak lower semi-continuity

: 10.1137/16M1060947


(eng): Minimization is a recurring theme in many mathematical disciplines ranging from pure\nto applied. Of particular importance is the minimization of integral functionals, which is\nstudied within the calculus of variations. Proofs of the existence of minimizers usually rely\non a fine property of the functional called weak lower semicontinuity. While early stud-\nies of lower semicontinuity go back to the beginning of the 20th century, the milestones\nof the modern theory were established by C. B. Morrey, Jr. [Pacific J. Math., 2 (1952),\npp. 25–53] in 1952 and N. G. Meyers [Trans. Amer. Math. Soc., 119 (1965), pp. 125–149]\nin 1965. We recapitulate the development of this topic from these papers onwards. Spe-\ncial attention is paid to signed integrands and to applications in continuum mechanics\nof solids. In particular, we review the concept of polyconvexity and special properties of\n(sub-)determinants with respect to weak lower semicontinuity. In addition, we empha-\nsize some recent progress in lower semicontinuity of functionals along sequences satisfying\ndifferential and algebraic constraints that can be used in elasticity to ensure injectivity\nand orientation-preservation of deformations. Finally, we outline generalizations of these\nresults to more general first-order partial differential operators and make some suggestions\nfor further reading

: BA

: 10101