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On the continuity of Jacobian of orientation preserving mappings in the grand Sobolev space. (English) Zbl 1299.26029

Let \(\Omega \) be a bounded domain in \(\mathbb R^n\) and let \(f=(f_1,\dots ,f_n)\: \Omega \to \mathbb R^n\) be a locally integrable mapping with a locally integrable distributional differential \(Df\). Its Jacobian determinant \(J_f(x)=\det Df(x)=\det (\partial f^i/\partial x_j)(x)\) is defined for almost every \(x\in \Omega \). The mapping \(f\) is orientation-preserving if \(J_f(x)\geq 0\) almost everywhere in \(\Omega \). The grand Lebesgue space \(L^{q)}(\Omega)\) for \(q>1\) is the class of all measurable functions \(f\) with the finite norm \(\| f\|_{L^{q)}}=\sup_{0<\varepsilon \leq q-1}\left (\varepsilon \int | f(x)| ^{q-\varepsilon }\right )^{1/(q-\varepsilon )}<\infty \) and the grand Sobolev space \(W^{1,q)}(\Omega)\) is the class of all functions \(f\in \bigcap_{0<\varepsilon \leq q-1}W^{1,q-\varepsilon }(\Omega)\) such that \(Df\in L^{q)}(\Omega)\), equipped with the norm \(\| f\|_{W^{1,q)}}=\| Df\|_{L^{q)}}+\| f\|_{L^{q)}}\). The authors prove that if \(f_j\) and \(f\) are orientation-preserving mappings in \(W^{1,q)}(\Omega ,\mathbb R^n)\) such that \(f_j\to f\) in \(W^{1,q)}(\Omega ,\mathbb R^n)\), then \(J_{f_j}\to J_f\) in \(L^1_{\text{loc}}(\Omega)\).

MSC:

26B10 Implicit function theorems, Jacobians, transformations with several variables
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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