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Higher integrability of determinants and weak convergence in \(L^ 1\). (English) Zbl 0713.49004
The following surprising higher integrability property of mappings u: \(\Omega \subset {\mathbb{R}}^ n\), \(n\geq 2\) with positive Jacobian is established: Assume that u is in the Sobolev space \(W^{1,n}\) and that det Du\(\geq 0\), then det Du ln(2\(+\det Du)\) is integrable. This generalizes Gehring’s results on quasiregular mappings, i.e. those satisfying det Du\(\geq c| Du|^ n\). Various extensions, including a “reverse-Hölder” type inequality are also obtained. The proof relies on a version of the isoperimetric inequality and a lemma of E. Stein on maximal functions in \(L^ 1\). Some of the results were announced by the author in Bull. Am. Math. Soc., New Ser. 21, No.2, 245- 248 (1989; Zbl 0689.49006).
Reviewer: St.Müller

MSC:
49J10 Existence theories for free problems in two or more independent variables
74B20 Nonlinear elasticity
26B35 Special properties of functions of several variables, Hölder conditions, etc.
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