The natural best approximant in Orlicz spaces of Young measures (Q1881092)

After introducing the space \({\mathcal Y}\) of Young measures over \([0,1]\times {\mathbb R}^d\), a class of functions \(\Phi\) from \([0,1]\times {\mathbb R}^d\) into \({\mathbb R}\), called the class of integrable vector Young functions, measurable with respect to the first variable, convex with respect to the second one, satisfying \(\Phi(t,x)=0\) if and only if \(x=0\), and satisfying some other conditions, the author introduces the problem \(P_\Phi\) ~minimize~ \(\int_0^1 \int_{ {\mathbb R}^d} \Phi(t,x) \nu(t)\, dt\), when \(\nu \in C\). In this setting \(C\) is a subset of \({\mathcal Y}_\Phi\), the space of Young measures \(\nu\) for which \(\int_0^1 \int_{ {\mathbb R}^d} \Phi(t,x) \nu(t)\, dt\) is finite. For a given family \((\Phi_\varepsilon)_\varepsilon\) of integrable vector Young functions, converging in an appropriate way to \(\Phi_0\), another integrable vector Young function, and for a sequence \(\mu_\varepsilon\), where \(\mu_\varepsilon\) is a minimizer of \(P_{\Phi_\varepsilon}\), the author proves that \((\mu_\varepsilon)_\varepsilon\) converges to \(\mu_0\), where \(\mu_0\) is the solution of \(P_{\Phi_0}\) of minimal \(\Theta\)-norm, where \(\Theta\) is an integrable vector Young function which is related in the manner in which \((\Phi_\varepsilon)_\varepsilon\) converges to \(\Phi_0\).