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Integrals with respect to a Radon measure added to area type functionals: semi-continuity and relaxation. (English) Zbl 0632.49005

In this expository note the authors present some results for integral functionals of the type \[ F(u)=\int_{\Omega}f(x,u,Du)dx+\int_{{\bar \Omega}}g(x,u)d\mu I=I(u)+\int_{\Omega}g(x,u)d\mu. \] In particular, under the assumption that I is \(L_ 1(\Omega)\)-lower semicontinuous, they present a result on the \(L_ 1(\Omega)\)-lower semicontinuity of F over \(W^{1,1}(\Omega)\cap C^ 0({\bar \Omega})\) and a result on the integral representation of the relaxed functional in the following terms: \(sc^-(L_ 1(\Omega))F=I(u)+\int_{\Omega}\gamma (x,u)d\mu\).
Reviewer: A.Salvadori

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
28C05 Integration theory via linear functionals (Radon measures, Daniell integrals, etc.), representing set functions and measures
49M20 Numerical methods of relaxation type
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