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Weak continuity and lower semicontinuity results for determinants. (English) Zbl 1081.49013

Summary: Weak continuity properties of minors and lower semicontinuity properties of functionals with polyconvex integrands are addressed in this paper. In particular, it is shown that if \(\{u_n\}\) is bounded in \(W^{1,N-1}(\Omega;\mathbb R^N)\), \(\{\text{adj}\nabla u_n\} \subset L^{\frac{N}{N-1}} (\Omega;\mathbb R^{N\times N})\), and if \(u \in \text{BV}(\Omega;\mathbb R^N)\) are such that \(u_n \rightarrow u\) in \(L^1(\Omega;\mathbb R^N)\) and \[ \det \nabla u_n \overset {*} \rightharpoonup \mu \] in the sense of measures, then for \(\mathcal L^N\) a.e. \(x \in \Omega\), \[ \det\nabla u(x) = \frac{d\mu}{d\mathcal L^N}(x). \] This result is sharp, and counterexamples are provided in the cases where the regularity of \(\{u_n\}\) or the type of weak convergence is weakened.

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
49N60 Regularity of solutions in optimal control
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