Carnegie Mellon
Department of Mathematical 

Martin Kruzik, Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic

"{\cal A}-quasiconvexity and lower semicontinuity of integral functionals "


Young measures proved to be an efficient tool to record statistical distributions of fast spatial oscillations in sequences of gradients living in a Lebesgue space. They are a key ingredient in mathematical models of shape-memory materials and modern calculus of variations. Concentration effects related to the lack of equiintegrability of the power of the gradient modulus however, cannot be treated by Young measures.We introduce a suitable generalization of Young measures, called DiPerna-Majda measures and characterize all these measures generated by sequences of Sobolev mappings $\{u_k\}$ satisfying a constraint ${\cal A}u_k=0$ where ${\cal A}$ is a first-order and constant rank partial differential operator. This result will be then used to obtain new sequential weak lower semicontinuity theorems for integral functionals along sequences of ${\cal A}$-free mappings. For a particular case ${\cal A}={\rm curl}$, we recover and generalize results related to weak sequential continuity of determinants due to S.~M\"{u}ller. This is mostly a joint work with Irene Fonseca.

TUESDAY, September 25, 2007
Time: 1:30 P.M.
Location: PPB 300