Oscillations and concentrations generated by $𝒜$-free mappings and weak lower semicontinuity of integral functionals

Fonseca, Irene; Kružík, Martin

ESAIM: Control, Optimisation and Calculus of Variations, Tome 16 (2010) no. 2, pp. 472-502.

DiPerna’s and Majda’s generalization of Young measures is used to describe oscillations and concentrations in sequences of maps ${\left\{u_k\right\}}_{k\in \mathbb{N}}\subset L^p(\Omega ;{ℝ}^m)$ satisfying a linear differential constraint $𝒜u_k=0$. Applications to sequential weak lower semicontinuity of integral functionals on $𝒜$-free sequences and to weak continuity of determinants are given. In particular, we state necessary and sufficient conditions for weak* convergence of det$\nabla {\varphi }_k\stackrel{*}{\rightharpoonup }\det \nabla \varphi$ in measures on the closure of $\Omega \subset {ℝ}^n$ if ${\varphi }_k\rightharpoonup \varphi$ in $W^{1,n}(\Omega ;{ℝ}^n)$. 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 $u\mapsto {\int }_{\Omega \setminus {\Omega }_j}v\left(u\left(x\right)\right)\mathrm{d}x$ along $\left\{u_k\right\}\subset L^p(\Omega ;{ℝ}^m)\cap \ker 𝒜$. Specifically, ${\Omega }_j$ are arbitrarily thin “boundary layers”.

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