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New lower semicontinuity results for nonconvex functionals defined on measures. (English) Zbl 0736.49007

Let \((\Omega,d)\) be a separable locally compact metric space and let \(\mu\) be a fixed positive, finite, non atomic measure. The authors consider a wide class of functionals defined on the space \({\mathcal M}(\Omega,R^ n)\) of vector valued measures with finite variation in \(\Omega\) and they prove a lower semicontinuity result with respect to the weak* convergence. The functionals have the form \[ F(\lambda)=\int_ \Omega f\left(x,{d\lambda\over d\mu}\right)+\int_{\Omega\backslash A_ \lambda}\varphi\left(x,{d\lambda^ s\over d|\lambda^ s|}\right)d|\lambda^ s|+\int_{A_ \lambda}g(x,\lambda(\{ x\}))d\# \] where \(A_ \lambda\) is the set of atoms of \(\lambda\), \(\lambda={d\lambda\over d\mu}+\lambda^ s\) is the Lebesgue-Nikodym decomposition of \(\lambda\), \(|\lambda|\) is the variation of \(\lambda\) and \(\#\) is the counting measure. Under the hypotheses considered in Theorem 3.2 the functional \(F\) is lower semicontinuous without being convex.
Reviewer: A.Leaci (Lecce)

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
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[1] Ambrosio, L.; Buttazzo, G., Weak lower semicontinuous envelope of functions defined on a space of measures, Ann. Mat. pura appl., 150, 311-340 (1988) · Zbl 0648.49009
[2] Bouchitté, G., Représentation intégrale de fonctionnelles convexes sur un espace de measures, Ann. Univ. Ferrara, 33, 113-156 (1987) · Zbl 0721.49041
[3] Bouchitté, G.; Valadier, M., Integral representation of convex functionals on a space of measures, J. funct. Analysis, 80, 398-420 (1988) · Zbl 0662.46009
[5] Buttazzo, G., Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations, (Pitman Research Notes in Mathematics, 207 (1989), Longman: Longman Harlow) · Zbl 0669.49005
[6] Buttazzo, G.; Dal, Maso G., On Nemyckii operators and integral representation of local functionals, Rc. Mat., 3, 491-509 (1983) · Zbl 0536.47027
[7] Cohn, D. L., Measure Theory (1980), Birkhäuser: Birkhäuser Basel · Zbl 0436.28001
[10] De Giori, E.; Ambrosio, L.; Buttazzo, G., Integral representation and relaxation for functionals defined on measures, Atti Accad. naz. Lincei Rc., 81, 7-13 (1987) · Zbl 0713.49018
[11] Goffman, C.; Serrin, J., Sublinear functions of measures and variational integrals, Duke math J., 31, 159-178 (1964) · Zbl 0123.09804
[12] Kuczma, M., An Introduction to the Theory of Functional Equations and Inequalities (1985), Panstwowe Wydawnictwo Naukowe: Panstwowe Wydawnictwo Naukowe Warszawa
[13] Laurent, P. J., Approximation et Optimisation (1972), Hermann: Hermann Paris · Zbl 0238.90058
[14] Rogers, C. A.; Jayne, J. E., K-analytic sets, (Rogers, C. A.; etal., Analytic Sets (1980), Academic Press: Academic Press New York) · Zbl 0524.54028
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