Explicit characterization of \(L^ p\)-Young measures
Let \(\Omega\) be an open domain, \({\mathcal Y}(\Omega,\mathbb{R}^m)\) be the set of Young measures. Let \(1\leq p<+\infty\), \((u_k)_k\) be a bounded sequence in \(L^p(\Omega,\mathbb{R}^m)\); then classically it admits a subsequence \((u_{k_j})_j\), and a Young measure \(\nu=(\nu_x)_{x\in\Omega}\in{\mathcal Y}(\Omega,\mathbb{R}^m)\) such that \[ \lim_{j\to\infty} v\circ u_{k_j}(x)= \int_{\mathbb{R}^m} v(\lambda)\nu_x d\lambda \] weakly in \(L^1(\Omega)\) for \(p<\infty\), weakly\(^*\) in \(L^\infty(\Omega)\) for \(p=\infty\). Consider the subclasses \({\mathcal Y}^p(\Omega,\mathbb{R}^m)\) of Young measures which are created in this way. An explicit characterization of \({\mathcal Y}^p(\Omega,\mathbb{R}^m)\) was already known only for \(p=\infty\). The main result of this paper characterizes \({\mathcal Y}^p(\Omega,\mathbb{R}^m)\) for \(1\leq p<\infty\) via the elegant equivalent condition \[ \Biggl(x\mapsto \int_{\mathbb{R}^m}|\lambda|^p \nu_xd\lambda\Biggr)\in L^1(\Omega). \] Next, the authors apply this result to the investigation of the Di Perna-Majda measures which are, according to the authors' introduction, a crucial \textit{tool to handle both oscillation and concentration effects simultaneously} in nonlinear problems. Briefly Di Perna-Majda measures are a class of pairs \((\sigma,\widehat\nu)\), where \(\sigma\in rca(\overline\Omega)\) and \(\widehat\nu\) is a Young measure linked to \(\sigma\) by a limit relation taking into account test continuous functions from a suitable subring of \({\mathcal C}(\mathbb{R}^m)\). The authors characterize such measures in special cases in terms of the absolute continuity of \(\sigma\) with respect to the Lebesgue measure \(dx\) and of the relationship between \({d\sigma\over dx}\) and \(\widehat\nu_x\).
scientific article; zbMATH DE number 906456