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Strong convergence results related to strict convexity. (English) Zbl 0545.49019

Let (\(\Omega\),\({\mathcal A},\mu)\) be a \(\sigma\)-finite, complete measure space. In section 1, the author proves the following result and gives some corollaries: Assume that \(u_ n\to u_ 0\) weakly in \(L^ 1(\Omega)^ N\) and \(u_ 0(x)\) is an extremal point of \(\overline{conv}(\cup^{\infty}_{n=0}\{u_ n(x)\})\) a.e. for \(x\in \Omega\). Then \(u_ n\to u_ 0\) strongly in \(L^ 1(\Omega)^ N\). In section 2, the author gives some applications to convex normal integrands \(\Phi\). Let \(u,\quad u_ n: \Omega \to {\mathbb{R}}\quad(n\in {\mathbb{N}})\) be measurable functions. Assume that (u(x),\(\Phi\) (u(x))) is an extremal point of epi \(\Phi\) (x) a.e. for \(x\in \Omega\). (i) If \(u_ n\to u\) weakly in \(L^ 1(\Omega)^ N\) and \(\Phi(u_ n)\to \Phi(u)\) weakly in \(L^ 1(\Omega)\), then \(u_ n\to u\) strongly in \(L^ 1(\Omega)^ N\) and \(\Phi(u_ n)\to \Phi(u)\) strongly in \(L^ 1(\Omega)\). (ii) Let \(1<p<+\infty\). If \(u_ n\to u\) weakly in \(L^ p_{loc}(\Omega)^ N\) and \(\Phi(u_ n)\to \Phi(u)\) weakly in \(L^ p_{loc}(\Omega)\), then, for \(1<q<p\), \(u_ n\to u\) strongly in \(L^ q_{loc}(\Omega)^ N\) and \(\Phi(u_ n)\to \Phi(u)\) strongly in \(L^ q_{loc}(\Omega)\). Some applications are presented, namely the applications of the preceding results to some differential equations and minimization problems.
Reviewer: C.Castaing

MSC:

49Q15 Geometric measure and integration theory, integral and normal currents in optimization
49J45 Methods involving semicontinuity and convergence; relaxation
52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
90C25 Convex programming
28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
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References:

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