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)


49J45 Methods involving semicontinuity and convergence; relaxation
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