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On the integrability of the Jacobian under minimal hypotheses. (English) Zbl 0766.46016
Let \(f=(f^ 1,f^ 2,\dots,f^ n):\Omega\to R^ n\), where \(\Omega\) is a region in \(R^ n\) and \(J(x,f)=\text{det} Df(x)\) the Jacobian of \(f\). The authors prove a number of results on integrability of \(J(x,f)\) under minimal assumptions on \(f\). These results start from Miller’s observation that the assumption that \(J(x,f)\) doesn’t change sign, implies higher integrability of \(J(x,f)\) in comparison with that of \(| Df(x)|^ n\). This observation is essentially developed and strengthened. In particular, it is shown that if \(J(x,f)\geq 0\) is a mapping of the Sobolev-Orlicz class \(D^ n\log^{-1}D\), then \[ \int_ EJ(x,f)dx\leq c(n,D)\int_ \Omega{| Df(x)|^ ndx\over\bigl|\log\bigl(e+{(Df(x))\over| Df|_ \Omega}\bigr)\bigr|} \] for each compact subset \(E\subset\Omega\), \(| Df|_ \Omega\) being the integral mean of \(| Df|\) over \(\Omega\).

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