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On the singular support of the distributional determinant. (English) Zbl 0792.46027
Summary: Let \(\Omega\subset \mathbb{R}^ n\) be bounded and open, let \(p\geq n^ 2/(n+1)\) and let \(u: \Omega\to\mathbb{R}^ n\) be in the Sobolev space \(W^{1,n}(\Omega;\mathbb{R}^ n)\). This paper discusses the singular part of the distributional determinant \(\text{Det }Du\) and shows the existence of functions \(u\) for which that singular part is supported in a set of prescribed Hausdorff-dimension \(\alpha\in(0,n)\). For \(n=2\) and simply connected \(\Omega\) the problem is equivalent to analyzing \(\text{div}(bv)- b.Dv\), where \(v\in W^{1,p}(\Omega;\mathbb{R}^ 2)\) with \(\text{div }b= 0\).

MSC:
46F10 Operations with distributions and generalized functions
26B10 Implicit function theorems, Jacobians, transformations with several variables
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