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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.

##### MSC:
 49J45 Methods involving semicontinuity and convergence; relaxation
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##### References:
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