Integral representation and relaxation for functionals defined on measures

De Giorgi, Ennio ; Ambrosio, Luigi ; Buttazzo, Giuseppe

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti, Tome 81 (1987), p. 7-13 / Harvested from Biblioteca Digitale Italiana di Matematica

Abstract

Given a separable metric locally compact space $\Omega$, and a positive finite non-atomic measure $\lambda$ on $\Omega$, we study the integral representation on the space of measures with bounded variation $\Omega$ of the lower semicontinuous envelope of the functional $$F(u) = \int_{\Omega} f(x,u) d\lambda \qquad u \in L^{1}(\Omega,\lambda,\mathbb{R}^{n})$$ with respect to the weak convergence of measures.

Résumé

Dato uno spazio metrico localmente compatto a base numerabile $\Omega$ ed una misura $\lambda$ su tale spazio, positiva, finita e non atomica, si studia la rappresentazione integrale del funzionale ottenuto rilassando $$F(u) = \int_{\Omega} f(x,u) d\lambda \qquad u \in L^{1}(\Omega,\lambda,\mathbb{R}^{n})$$ nello spazio $M_{n}(\Omega)$ delle misure a variazione limitata su $\Omega$, rispetto alla topologia della convergenza debole di misure.

Détail

Publié le : 1987-03-01

MR 1000018 | Zbl 0713.49018

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