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On the optimality of certain Sobolev exponents for the weak continuity of determinants. (English) Zbl 0769.46025
Suppose that \(u_ \varepsilon\) and \(u\) are elements in \(W^{1,n}(\Omega;\mathbb{R}^ n)\) such that \(u_ \varepsilon\to u\) weakly in \(W^{1,p}(\Omega;\mathbb{R}^ n)\). If \(p>n^ 2/(n+1)\), then it is known that \(\text{det }\nabla u_ \varepsilon\) converges to \(\text{det }\nabla u\) weakly in \({\mathcal D}'(\Omega)\). The aim of the paper under review is to show that this sequential continuity fails to hold if \(1\leq p\leq n^ 2/(n+1)\). More precisely, if \(p=n^ 2/(n+1)\), then a subsequence of \(\text{det }\nabla u_ \varepsilon\) converges weakly in \({\mathcal D}'(\Omega)\) to \(\text{det } \nabla u+\mu\), where \(\mu\) is a Radon measure, nonzero in general. For \(1\leq p<n^ 2/(n+1)\) the authors find \(u_ \varepsilon\) and \(u\) as above and \(\varphi\in{\mathcal D}(\Omega)\) such that \[ \int_ \Omega \text{det } \nabla u_ \varepsilon \varphi dx\to\infty. \] They also prove analogous results for \(\text{det }^ 2 \nabla v_ \varepsilon\) that converges in \(W^{2,p}(\Omega; \mathbb{R}^ n)\); then the critical value of \(p\) is \(p=n^ 2/(n+2)\).

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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