Weak lower semicontinuous envelope of functionals defined on a space of measures. (English) Zbl 0648.49009

Let \((\Omega,{\mathcal B},\lambda)\) be a Borel measure space on which the measure \(\lambda\) is positive and nonatomic, with \(\lambda (\Omega)<+\infty\). Consider the functional \[ F(u):=\int^{*}_{\Omega}f(x,u(x))dx,\quad u\in L_ 1(\Omega,{\mathcal B},\lambda;{\mathbb{R}}^ n)\equiv L^ n_ 1, \] where \(\int^{*}\) denotes the Lebesgue upper integral and \(f: \Omega\times {\mathbb{R}}^ n\to [0,+\infty]\) is a given extended valfued function subject to no measurability hypotheses. The reference measure \(\lambda\) sponsors a natural embedding of \(L^ n_ 1\) in the space \(M^ n\) of \({\mathbb{R}}^ n\)-valued measures on \((\Omega,{\mathcal B})\) with finite variation, namely \(u\mapsto \tilde u\), where \(d\tilde u(x)=u(x)d\lambda (x)\). The natural extension of F to the domain \(M^ n\) is given by \(\tilde F(\tilde u)=F(u)\) if \(\tilde u\in \tilde L^ n_ 1\), \(\tilde F(\tilde u)=+\infty\) otherwise. Now let \({\tilde \Phi}\) be the greatest sequentially weak*-lower semicontinuous functional on \(M^ n\) majorized by \(\tilde F\). This paper provides a tidy integral representation of \({\tilde \Phi}\), thus solving the “relaxation problem” described so beautifully in the introduction. This formula is derived from a more general integral representation theorem pertaining to extended-valued functionals on \(M^ n\) which share certain key properties with the \(\tilde F\) described above.
Reviewer: P.Loewen


49J45 Methods involving semicontinuity and convergence; relaxation
28A33 Spaces of measures, convergence of measures
54C08 Weak and generalized continuity
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[1] Acerbi, E.; Fusco, N., Semicontinuity problems in the calculus of variations, Arch. Rational Mech. Anal., 86, 125-145 (1984) · Zbl 0565.49010
[2] G.Anzellotti - G.Buttazzo - G. DalMaso,Dirichlet problems for demi-coercive functionals, Nonlinear Anal., to appear. · Zbl 0612.49008
[3] G.Bouchitte, Paper in preparation.
[4] Buttazzo, G.; Dal Maso, G., On Nemyckii operators and integral representation of local functionals, Rend. Mat., 3, 491-509 (1983) · Zbl 0536.47027
[5] Buttazzo, G.; Dal Maso, G., Integral representation and relaxation of local functionals, Nonlinear Anal., 9, 515-532 (1985) · Zbl 0527.49008
[6] De Giorgi, E.; Letta, G., Une notion générale de convergence faible pour des fonctions croissantes d’ensemble, Ann. Scuola Norm. Sup. Pisa, Cl. Sci., 4, 61-99 (1977) · Zbl 0405.28008
[7] E. DeGiorgi - L.Ambrosio - G.Buttazzo,Integral representation and relaxation for functionals defined on measures, Atti Accad. Naz. dei Lincei, Rend. Cl. Sci. Fis. Mat. Natur., to appear. · Zbl 0713.49018
[8] De Giorgi, E.; Buttazzo, G.; Dal Maso, G., On the lower semicontinuity of certain integral functionals, Atti Accad. Naz. dei Lincei, Rend. Cl. Sci. Fis. Mat. Natur., 74, 274-282 (1983) · Zbl 0554.49006
[9] Ekeland, I.; Temam, R., Convex Analysis and Variational Problems (1976), Amsterdam: North Holland, Amsterdam · Zbl 0322.90046
[10] Federer, H., Geometric Measure Theory (1969), Berlin: Springer-Verlag, Berlin · Zbl 0176.00801
[11] A.Fougeres - A.Truffert,A-integrands and essential infimum, Nemyckii representation of l.s.c. operators on decomposable spaces and Radon-Nikodym-Hiai representation of measure functionals, Preprint AVAMAC University of Perpignan, Perpignan, 1984.
[12] A.Gavioli,Condizioni necessarie e sufficienti per la semicontinuità inferiore di certi fun- zionali integrali, Preprint University of Modena, Modena, 1986.
[13] Halmos, P. R., Measure Theory (1950), Princeton: Van Nostrand, Princeton · Zbl 0040.16802
[14] Hiai, F., Representation of additive functionals on vector valued normed Kothe spaces, Kodai Math. J., 2, 300-313 (1979) · Zbl 0431.46025
[15] Marcellini, P.; De Giorgi, E.; Magenes, E.; Mosco, U., Some problems of semicontinuity, Proceedings «Recent Methods in Non- linear Analysis», Rome, 1978, 205-222 (1979), Bologna: Pitagora, Bologna
[16] Marcellini, P.; Sbordone, C., Semicontinuity problems in the calculus of variations, Nonlinear Anal., 4, 241-257 (1980) · Zbl 0537.49002
[17] Olech, C.; Antosiewicz, H. A., Existence theory in optimal control problems: the underlying ideas, Proceedings of «International Conference on Differential Equations», University of Southern California, 1974, 612-629 (1975), New York: Academic Press, New York · Zbl 0353.49013
[18] Rockafellar, R. T., Integrals which are convex functionals, I, Pacific J. Math., 39, 439-469 (1971) · Zbl 0236.46031
[19] Rockafellar, R. T., Convex Analysis (1972), Princeton: Princeton University Press, Princeton · Zbl 0224.49003
[20] Rudin, W., Real and Complex Analysis (1974), New York: Mc-Graw Hill, New York · Zbl 0278.26001
[21] Valadier, M., Closedness in the weak topology of the dual pair L^1,C, J. Math. Anal. Appl., 69, 17-34 (1979) · Zbl 0412.46040
[22] Valadier, M., Some bang-bang theorems, Lecture Notes in Math.1091 (1984), Berlin: Springer-Verlag, Berlin · Zbl 0562.49006
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