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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 $$p=n^ 2/(n+2)$$.

MSC:
 4.6e+36 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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References:
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