Weak lower semicontinuity of integral functionals and applications

TL;DR

This paper reviews the development of weak lower semicontinuity of integral functionals in calculus of variations, highlighting key concepts like polyconvexity, applications in continuum mechanics, and recent advances involving differential constraints.

Contribution

It provides a comprehensive overview of the historical and recent progress in weak lower semicontinuity, especially in relation to signed integrands and applications in elasticity.

Findings

Review of classical and modern theories of lower semicontinuity.

Discussion of polyconvexity and determinant properties.

Recent progress on functionals with differential constraints in elasticity.

Abstract

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.