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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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##### References:
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