Integral representation of nonconvex functionals defined on measures. (English) Zbl 0757.49012

The paper deals with an integral representation problem for nonconvex functionals defined on the space \(M(\Omega;\mathbb{R}^ n)\) of the \(\mathbb{R}^ n\)-valued measures with bounded variation on \(\Omega\). More precisely given a separable locally convex metric space \(\Omega\) and a functional \(F:M(\Omega;\mathbb{R}^ n)\to[0,+\infty]\) it is proved that if \(F\) verifies the following assumptions: (i) \(F\) is additive (i.e. \(F(\lambda+\mu)=F(\lambda)+F(\mu)\) whenever \(\lambda,\mu\in M(\Omega;\mathbb{R}^ n)\) and \(\lambda\) is singular with respect to \(\mu)\), (ii) \(F\) is sequentially weakly* lower semicontinuous on \(M(\Omega;\mathbb{R}^ n)\) then there exist a non-atomic positive measure \(\mu\) in \(M(\Omega;\mathbb{R}^ n)\) and three Borel functions \(f,\varphi,g:\Omega\times\mathbb{R}^ n\to[0,+\infty]\) verifying suitable conditions such that the following integral representation formula holds for every \(\lambda\in M(\Omega;\mathbb{R}^ n)\): \[ F(\lambda)=\int_ \Omega f\left(x,{d\lambda\over d\mu}\right)d\mu+\int_{\Omega\backslash A_ \lambda}\varphi\left(x,{d\lambda^ 2\over d|\lambda^ s|}\right)d|\lambda^ s|+\int_{A_ \lambda}g(x,\lambda(\{ x\}))d\#, \tag{*} \] where \({d\lambda\over d\mu}\) denotes the Radon- Nikodym derivative of \(\lambda\) with respect to \(\mu\), \({d\lambda^ s\over d|\lambda^ s|}\) the one of \(\lambda^ s\) with respect to \(|\lambda^ s|\), \(A_ \lambda\) is the set of the atoms of \(\lambda\) and \(\#\) is the counting measure on \(\Omega\). The uniqueness of the representation of \(F\) in the form \((*)\) is also discussed.
The above result extends to the nonconvex case analogous integral representation theorems already existing in literature but relative to convex functionals verifying (i) and (ii).
It is also proved that the necessary conditions on \(f\), \(\varphi\) and \(g\) given by the previous theorem, together with a mild additional assumption, become sufficient in order to get a weak*-\(M(\Omega;\mathbb{R}^ n)\) lower semicontinuity result for the functional in \((*)\). This result extends those already existing in literature.
Finally some examples are discussed, in particular the last one proves that the assumptions of the abover mentioned lower semicontinuity result are sharp.


49J45 Methods involving semicontinuity and convergence; relaxation
Full Text: DOI Numdam EuDML


[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, 7, 679-692 (1990) · Zbl 0736.49007
[4] 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
[5] Bouchitté, G.; Valadier, M., Multifonctions s.c.i. et régularisée s.c.i. essentielle. Proceedings “Congrès Franco-Québécois d’Analyse Non linéaire Appliquée”, Perpignan, June 22-26, 1987, Analyse Non linéaire - Contribution en l’Honneur de J. J. Moreau, 123-149 (1989), Bordas: Bordas Paris · Zbl 0704.49017
[6] 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
[7] Buttazzo, G.; Dal Maso, G., On Nemyckii operators and integral representation of local functionals, Rend. Mat., Vol. 3, 491-509 (1983) · Zbl 0536.47027
[8] de Giorgi, E.; Ambrosio, L.; Buttazzo, G., Integral representation and relaxation for functionals defined on measures, Atti Accad. Naz. Lincei Rend. CI Sci. Fis. Mat. Natur., Vol. 81, 7-13 (1987) · Zbl 0713.49018
[9] Demengel, F.; Теmam, R., Convex functions of measures and applications, Indiana Univ. Math. J., 33, 673-709 (1984) · Zbl 0581.46036
[10] Dunford, N.; Schwartz, J. T., Linear Operators (1957), Interscience Publishers Inc: Interscience Publishers Inc New York
[11] Goffman, C.; Serrín, J., Sublinear functions of measures and variational integrals, Duke Math. J., Vol. 31, 159-178 (1964) · Zbl 0123.09804
[12] Hiai, F., Representation of additive functionals on vector valued normed Köthe spaces, Kodai Math. J., Vol. 2, 300-313 (1979) · Zbl 0431.46025
[13] Ioffe, A. D., On lower semicontinuity of integral functionals I. II, SIAM J. Control Optim., Vol. 15, 521-538 (1977), nd 991-1000 · Zbl 0361.46037
[14] Rockafellar, R. T., Integrals which are convex functionals I. II, Pacific J. Math., Vol. 39, 439-469 (1971) · Zbl 0236.46031
[15] Rockafellar, R. T., Convex Analysis (1972), Princeton University Press: Princeton University Press Princeton · Zbl 0224.49003
[16] Valadier, M., Closedness in the weak topology of the dual pair \(L^1, C\), J. Math. Anal. Appli., Vol. 69, 17-34 (1979) · Zbl 0412.46040
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.