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Lower semicontinuity and relaxation for extended-valued integrands

Mikhail Sychev
Tuesday, April 24, 2018 - 16:00

We discuss various methods of establishing lower semicontinuity of integral functionals. First we show that if original integrand $L$ can be approximated locally uniformly by integrands $L_k$ with $p$ -growth such that ($L_k +\Phi)^qc$ converges pointwisely to $(L + \Phi)^qc$ for each $C^{\infty}$-regular integrand $\Phi$ with compact support then lower semicontinuity holds in biting sense for the integral functional with the integrand $L$. We isolate condition which is both necessary and sufficient for lower semicontinuity of $p$-coercive problems: $p$-quasiconvexity and condition (M). Condition (M) means that when approximating linear functions by Sobolev functions the sequence can be replaced (up to subsequence) by functions with the linear boundary data without increase of energy in the limit. It turned out that there is rather simple relaxation theory in the case when condition (M) holds. 

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