Interchange of infimum and integral (Q1422211)

In the paper under review the authors prove the interchange formula \[ \inf_{w\in{\mathcal H}}\, \int_\Omega f(x,w(x))\,d\mu(x)= \int_\Omega\Biggl(\inf_{\xi\in\Gamma(x)}\, f(x,\xi)\Biggr)\,d\mu(x), \] where \({\mathcal H}\) is a subset of \(L^p_\mu(\Omega; X)\) and \(\Gamma: \Omega\to X\) is a multimapping. Here \(\Omega\) is a locally compact metric space and \(X\) is a real separable Banach space. The main assumption under which the interchange formula is proved is that the subset \({\mathcal H}\) is ``\textit{normally decomposable}'', that is for every \(u,v\in{\mathcal H}\), every open set \(V\subset\Omega\) and every compact set \(K\subset V\) there exists a Urysohn function \(\phi\) separating \(K\) from \(\Omega\setminus V\) such that \(\phi u+ (1-\phi)v\in{\mathcal H}\). The result generalizes the previous ones in the literature (by Rockafellar, Hiai and Umegaki, Bouchitté and Valadier), that are all recalled and discussed. The last section of the paper contains some applications to integral functionals of the calculus of variations of the form \[ \int f(x,\nabla u(x))\,d\mu(x). \]