Relaxation for a class of nonconvex functionals defined on measures. (English) Zbl 0791.49016

Summary: We characterize in a suitable integral form like \[ \overline F(\lambda)= \int_ \Omega\overline f\left(x, {d\lambda\over d\mu}\right) d\mu+ \int_{\Omega\backslash A_ \lambda} \overline\varphi(x,\lambda^ s)+ \int_{A_ \lambda} \overline g(x,\lambda(x))d\# \] the lower semicontinuous envelope \(\overline F\) of functionals \(F\) defined on the space \({\mathcal M}(\Omega;{\mathbf R}^ n)\) of all \({\mathbf R}^ n\)-valued measures with finite variation on \(\Omega\).


49J45 Methods involving semicontinuity and convergence; relaxation
46G10 Vector-valued measures and integration
46E27 Spaces of measures
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[1] Ambrosio, L.; Buttazzo, G., Weak Lower Semicontinuous Envelope of Functionals Defined on a Space of Measures, Ann. Mat. Pura Appl., Vol. 150, 311-340 (1988) · Zbl 0648.49009
[2] Bouchitté, G., Représentation intégrale de fonctionnelles convexes sur un espace de mesures, Ann. Univ. Ferrara, Vol. 33, 113-156 (1987) · Zbl 0721.49041
[3] Bouchitté, G.; Buttazzo, G., New Lower Semicontinuity Results for non Convex Functionals defined on Measures, Nonlinear Anal., Vol. 15, 679-692 (1990) · Zbl 0736.49007
[5] Bouchitté, G.; Valadier, M., Integral Representation of Convex Functionals on a Space of Measures, J. Funct. Anal., Vol. 80, 398-420 (1988) · Zbl 0662.46009
[7] Buttazzo, G., Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations, Pitman Res. Notes Math. Ser., Vol. 207 (1989), Longman: Longman Harlow · Zbl 0669.49005
[8] Buttazzo, G.; Maso, G. Dal, On Nemyckii Operators and Integral Representation of Local Functionals, Rend. Mat., Vol. 3, 491-509 (1983) · Zbl 0536.47027
[9] de Giorgi, E.; Ambrosio, L.; Buttazzo, G., Integral Representation and Relaxation for Functionals defined on Measures, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., Vol. 81, 7-13 (1987) · Zbl 0713.49018
[10] Demengel, F.; Temam, R., Convex Functions of Measures and Applications, Indiana Univ. Math. J., Vol. 33, 673-709 (1984) · Zbl 0581.46036
[11] Dunford, N.; Schwartz, J. T., Linear Operators (1957), Interscience Publishers Inc.: Interscience Publishers Inc. New York
[12] Goffman, C.; Serrin, J., Sublinear Functions of Measures and Variational Integrals, Duke Math. J., Vol. 31, 159-178 (1964) · Zbl 0123.09804
[13] Hiai, F., Representation of Additive Functionals on Vector Valued normed Kothe Spaces, Kodai Math. J., Vol. 2, 300-313 (1979) · Zbl 0431.46025
[14] Ioffe, A. D., On Lower Semicontinuity of Integral Functionals I. II, SIAM J. Control Optim., Vol. 15, 521-538 (1977), and 991-1000 · Zbl 0361.46037
[15] Rockafellar, R. T., Integrals which are Convex Functionals I. II, Pacific J. Math., Vol. 39, 439-469 (1971), and · Zbl 0236.46031
[16] Rockafellar, R. T., Convex Analysis (1972), Princeton University Press: Princeton University Press Princeton · Zbl 0224.49003
[17] Valadier, M., Closedness in the Weak Topology of the Dual Pair \(L^1, C\), J. Math. Anal. Appl., Vol. 69, 17-34 (1979) · Zbl 0412.46040
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