Oscillations and concentrations generated by 𝒜-free mappings and weak lower semicontinuity of integral functionals
ESAIM: Control, Optimisation and Calculus of Variations, Volume 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 {u k } k L p (Ω; 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 detϕ k * det ϕ in measures on the closure of Ω n if ϕ k ϕ in W 1,n (Ω; n ). This convergence holds, for example, under Dirichlet boundary conditions. Further, we formulate a Biting-like lemma precisely stating which subsets Ω j Ω must be removed to obtain weak lower semicontinuity of u ΩΩ j v(u(x))dx along {u k }L p (Ω; m ) ker 𝒜. Specifically, Ω j are arbitrarily thin “boundary layers”.

DOI: 10.1051/cocv/2009006
Classification: 49J45,  35B05
Keywords: concentrations, oscillations, Young measures
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Fonseca, Irene; Kružík, Martin. Oscillations and concentrations generated by ${\mathcal {A}}$-free mappings and weak lower semicontinuity of integral functionals. ESAIM: Control, Optimisation and Calculus of Variations, Volume 16 (2010) no. 2, pp. 472-502. doi : 10.1051/cocv/2009006. http://www.numdam.org/articles/10.1051/cocv/2009006/

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