Young measures generated by a class of integrands: A narrow epicontinuity and applications to homogenization.
Given a subset \(E\) of convex functions from \({\mathbb R}^{N+k}\) into \({\mathbb R}\) satisfying a growth condition of order \(p>1\), and an open bounded subset \(\Omega=\widehat\Omega\times\Omega^\bot\) of \({\mathbb R}^{N+k}\), a sequence of functions \(\{f_\varepsilon\}\) from \(\widehat\Omega\times{\mathbb R}\times{\mathbb R}^k\) to \({\mathbb R}\) such that \(f_\varepsilon(\hat x,\cdot,\cdot)\in E\) is considered, and it is assumed that the image \(\mu^\varepsilon\) of the Lebesgue measure on \(\widehat\Omega\) by the map \(\hat x\mapsto(\hat x,f_\varepsilon(\hat x,\cdot,\cdot))\) possesses a weak limit \(\mu\) on \(\widehat\Omega\times E\). Then, it is proved that the sequence of integral functionals defined in \(L^p(\widehat\Omega,W^{1,p}_0(\Omega^\bot))\) by \[ F_\varepsilon(u)=\int_\Omega f_\varepsilon(\hat x,u(x),D^\bot u(x))dx \] \(\Gamma\)-converges to a functional \(\Phi_\mu\) of the form \[ \Phi_\mu(u)=\inf\left\{\int_{\Omega\times E}\Lambda(U(x,\Lambda),D^\bot U(x,\Lambda))d\mu(\hat x,\Lambda)dx^\bot : U\in X_{u,\mu}\right\}, \] where the symbol \(D^\bot\) denotes the gradient operator with respect to \(x^\bot\in\Omega^\bot\), and \[ X_{u,\mu}=\left\{U\in L^p_\mu(\widehat\Omega\times E,W^{1,p}_0(\Omega^\bot)) : \int_E U(x,\Lambda)d\mu_{\hat x}(\Lambda)=u(x)\text{ for a.e. }x\in\Omega\right\}, \] and where \(\mu_{\hat x}\) is a probability measure on \(E\), stemming from a disintegration \((\mu_{\hat x})_{\hat x\in\widehat\Omega}\) of \(\mu\). The continuity of \(\mu\mapsto\Phi_\mu\) is established from the set of all Young measures on \(\Omega\times E\) equipped with the narrow topology into the set of all the functionals of the form \(\Phi_\mu\) equipped with the topology of \(\Gamma\)-convergence. Applications are given to periodic and stochastic homogenization.
scientific article; zbMATH DE number 1975972
scientific article