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Mathematics Colloquium: Lower semicontinuity of integral functionals02-03-2020

Speaker:  Martin Kruzik, Czech Academy of Sciences

Title: Lower semicontinuity of integral functionals 

Date/Time: 4 March, 2020  /  13.40-14.30

Place: FENS G035

Abstract: 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 more general first-order partial differential operators with applications to electromagnetism, for instance.