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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.
Zbl 0689.49006
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