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Publication 08-CNA-19

Oscillations and Concentrations Generated by $ {\cal A}$-free Mappings and Weak Low Semicontinuity of Integral Functions

Irene Fonseca
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
fonseca@andrew.cmu.edu

Martin Kruzík
Institute of Information Theory and
Automation of the ASCR
Pod vodárenskon vezí 4
CZ-182 08 Praha 8
Czech Republic
kruzik@utia.cas.cz

 

Abstract: DiPerna's and Majda's generalization of Young measures is used to describe oscillations and concentrations in sequences of maps $ \{u_k\}_{k \in \mathbb{N}} \subset L^p(\Omega; \mathbb{R}^m)$ satisfying a linear differential constraint $ {\cal A}{u_k}=0$. Applications to sequential weak lower semicontinuity of integral functionals on $ {\cal A}$-free sequences and to weak continuity of determinants are given. In particular, we state necessary and sufficient conditions for weak* convergence of det $ \nabla{\varphi_k} \overset{*}{\rightharpoonup}\ {\rm det} \nabla \varphi$ in measures on the closure of $ \Omega \subset \mathbb{R}^n$ if $ \varphi_k \rightharpoonup \varphi$ in $ W^{1,n}(\Omega;\mathbb{R}^n)$. This convergence holds, for example, under Dirichlet boundary conditions. Further we formulate a Biting-like lemma precisely stating which subsets $ \Omega_j \subset \Omega$ must be removed to obtain weak lower semicontinuity of $ u \mapsto \int_{\Omega\backslash \Omega_j}v(u(x)) dx$ along $ \{u_k\} \subset L^p(\Omega; \mathbb{R}^m) \cap$ ker $ {\cal A}$. Specifically, $ \Omega_j$ are arbitrarily thin ``boundary layers.[5 ''

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