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

De Giorgi, Ennio; Ambrosio, Luigi; Buttazzo, Giuseppe

Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 81, No. 1, 7-13 (1987).

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.

MSC:

49J45

Methods involving semicontinuity and convergence; relaxation

Keywords:

relaxation; ingegral representation; measures with bounded variation; lower semicontinuous envelope