Lecture Course 250 136 VU:
Time: January 10 - 19, 2022
Start: January 10, 2022, 9 - 12 am
Further dates: January 12, 14, 17, 2022, 9 - 12 am
End: January 19, 2022, 9 - 11 am
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.
Martin Kružík (https://www.utia.cas.cz/people/kruzik) is a leading figure in the Calculus of Variations community. His research focuses at the crossing point of Analysis and Materials Science. The field is currently experiencing a very rapid development, with a constant influx of new problems, ideas, and efforts. Martin Kružík has contributed in shaping this development through his results, which have been influential under different respects. In particular, his work on oscillations and concentrations and his contribution to the modeling of microstructure evolution, especially for magnetism, are a reference.
He is currently Director of the Department of Decision-Making Theory of the Institute of Information Theory and Automation of the Czech Academy of Sciences in Prague, and he is teaching at Czech Technical University.
|Martin Kružík||Czech Academy of Sciences|