Bouchitté, Guy; Buttazzo, Giuseppe New lower semicontinuity results for nonconvex functionals defined on measures. (English) Zbl 0736.49007 Nonlinear Anal., Theory Methods Appl. 15, No. 7, 679-692 (1990). 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) Cited in 1 ReviewCited in 37 Documents MSC: 49J45 Methods involving semicontinuity and convergence; relaxation Keywords:non-convex integrals; functionals on measures × Cite Format Result Cite Review PDF Full Text: DOI References: [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 · Zbl 0555.39004 [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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.