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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References:

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