Weak Lower Semicontinuity of Integral Functionals and Applications

0481321 - ÚTIA 2018 RIV US eng J - Journal Article
Benešová, B. - Kružík, Martin
Weak Lower Semicontinuity of Integral Functionals and Applications.
SIAM Review. Roč. 59, č. 4 (2017), s. 703-766. ISSN 0036-1445. E-ISSN 1095-7200
R&D Projects: GA ČR GA14-15264S; GA ČR(CZ) GF16-34894L
Grant - others:GA AV ČR(CZ) DAAD-16-14
Program: Bilaterální spolupráce
Institutional support: RVO:67985556
Keywords : calculus of variations * weak lower semi-continuity
OECD category: Pure mathematics
Impact factor: 4.886, year: 2017
http://library.utia.cas.cz/separaty/2017/MTR/kruzik-0481321.pdf

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 stud-
ies 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. Spe-
cial 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 empha-
size 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
Permanent Link: http://hdl.handle.net/11104/0277002