## New lower semicontinuity results for nonconvex functionals defined on measures.(English)Zbl 0736.49007

Let $$(\Omega,d)$$ be a separable locally compact metric space and let $$\mu$$ be a fixed positive, finite, non atomic measure. The authors consider a wide class of functionals defined on the space $${\mathcal M}(\Omega,R^ n)$$ of vector valued measures with finite variation in $$\Omega$$ and they prove a lower semicontinuity result with respect to the weak* convergence. The functionals have the form $F(\lambda)=\int_ \Omega f\left(x,{d\lambda\over d\mu}\right)+\int_{\Omega\backslash A_ \lambda}\varphi\left(x,{d\lambda^ s\over d|\lambda^ s|}\right)d|\lambda^ s|+\int_{A_ \lambda}g(x,\lambda(\{ x\}))d\#$ where $$A_ \lambda$$ is the set of atoms of $$\lambda$$, $$\lambda={d\lambda\over d\mu}+\lambda^ s$$ is the Lebesgue-Nikodym decomposition of $$\lambda$$, $$|\lambda|$$ is the variation of $$\lambda$$ and $$\#$$ is the counting measure. Under the hypotheses considered in Theorem 3.2 the functional $$F$$ is lower semicontinuous without being convex.
Reviewer: A.Leaci (Lecce)

### MSC:

 49J45 Methods involving semicontinuity and convergence; relaxation

### Keywords:

non-convex integrals; functionals on measures
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### References:

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