分野別セミナー

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