Document Zbl 0766.46016

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Ration. Mech. Anal.</a> <a href="/?q=in%3A10205" title="Articles in this Issue">119, No. 2, 129-143 (1992)</a>. </div> <div class="abstract">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 $\Omega$.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A1492">S.G.Samko (Rostov-na-Donu)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A96945">Cited in <strong>14</strong> Reviews</a></div><div class="clear"><a href="/?q=rf%3A96945">Cited in <strong>292</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A46E35" title="MSC2020">46E35</a> </td> <td class="space"> Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A26B10" title="MSC2020">26B10</a> </td> <td class="space"> Implicit function theorems, Jacobians, transformations with several variables </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3AJacobian">Jacobian</a>; <a href="/?q=ut%3ASobolev-Orlicz+class">Sobolev-Orlicz class</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 0766.46016" data-ciurl="/ci/00096945" data-biburl="/bibtex/00096945.bib" data-amsurl="/amsrefs/00096945.bib" data-xmlurl="/xml/00096945.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/00096945.pdf" title="Zbl 0766.46016 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1007/BF00375119" aria-label="DOI for “On the integrability of the Jacobian under minimal hypotheses”" title="10.1007/BF00375119">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Acerbi, E., & Fusco, N., Semicontinuity problems in the calculus of variations, Arch. 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