## 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

### Keywords:

Jacobian; Sobolev-Orlicz class
Full Text:

### References:

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