Martin Kružík - Lower Semicontinuity of Integral Functionals and Applications, Lecture 4

This talk was part of the of Senior Research Fellow Lecture Series on "Lower Semicontinuity of Integral Functionals and Applications" held at the ESI January 10 to 19, 2022.

In 1830, B. Bolzano observed that continuous functions attain extreme values on compact intervals of reals. This idea was then significantly extended around 1900 by D. Hilbert who set up a framework, called the direct method, in which we can prove existence of minimizers/maximizers of nonlinear functionals. Semicontinuity plays a crucial role in these considerations. In 1965, N.G. Meyers significantly extended lower semicontinuity results for integral functionals depending on maps and their gradients available at that time. We recapitulate the development on this topic from that time on. Special attention will be paid to applications in continuum mechanics of solids. In particular, we review existing results applicable in nonlinear elasticity and emphasize the key importance of convexity and subdeterminants of matrix-valued gradients. Finally, we mention a couple of open problems and outline various generalizations of these results to moregeneral first-order partial differential operators with applications to electromagnetism, for instance.

Видео Martin Kružík - Lower Semicontinuity of Integral Functionals and Applications, Lecture 4 канала Erwin Schrödinger International Institute for Mathematics and Physics (ESI)