Integral representation and relaxation for functionals defined on measures. (English) Zbl 0713.49018

Summary: Given a separable metric locally compact space \(\Omega\), and a positive finite non-atomic measure \(\lambda\) on \(\Omega\), we study the ingegral representation on the space of measures with bounded variation \(\Omega\) of the lower semicontinuous envelope of the functional \[ F(u)=\int_{\Omega}f(x,u)d\lambda \quad u\in L^ 1(\Omega,\lambda,{\mathbb{R}}^ n) \] with respect to the weak convergence of measures.


49J45 Methods involving semicontinuity and convergence; relaxation