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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

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
28A33 Spaces of measures, convergence of measures
54C08 Weak and generalized continuity
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