Document Zbl 0689.49006

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Am. Math. Soc., New Ser.</a> <a href="/?q=in%3A155317" title="Articles in this Issue">21, No. 2, 245-248 (1989)</a>. </div> <div class="abstract">Let $n>1$, let $\Omega$ be an open subset of ${\mathbb{R}}^ n$ and let u be in the Sobolev space $W^{1,n}(\Omega;{\mathbb{R}}^ n)$. It is shown that if det Du$\geq 0$ then $\det Du \log (2+\det Du)\in L^ 1loc(\Omega).$ The proof relies on a version of the isoperimetric inequality and on a lemma on maximal functions in $L^ 1$ by <span class="zbmathjax-textit">E. Stein</span> [Studia Math. 32, 305-310 (1969; Zbl 0182.478)]. Applications to nonlinear elasticity are indicated. <br class="zbmathjax-paragraph">Subsequently new proofs and various generalizations of the result have been obtained by <span class="zbmathjax-textit">R. Coifman</span>, <span class="zbmathjax-textit">P. L. Lions</span>, <span class="zbmathjax-textit">Y. Meyer</span> and <span class="zbmathjax-textit">S. Semmes</span> [C. R. Acad. Sci., Paris, Ser. I 309, No.18, 945-949 (1989)].<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A6872">Stefan Müller</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A4129491">Cited in <strong>4</strong> Reviews</a></div><div class="clear"><a href="/?q=rf%3A4129491">Cited in <strong>28</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A49J20" title="MSC2020">49J20</a> </td> <td class="space"> Existence theories for optimal control problems involving partial differential equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A58C25" title="MSC2020">58C25</a> </td> <td class="space"> Differentiable maps on manifolds </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A74B20" title="MSC2020">74B20</a> </td> <td class="space"> Nonlinear elasticity </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A26B35" title="MSC2020">26B35</a> </td> <td class="space"> Special properties of functions of several variables, Hölder conditions, etc. </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Ahigher+integrability">higher integrability</a>; <a href="/?q=ut%3Aweak+continuity">weak continuity</a>; <a href="/?q=ut%3Aconcentration+compactness">concentration compactness</a>; <a href="/?q=ut%3Aisoperimetric+inequality">isoperimetric inequality</a>; <a href="/?q=ut%3Anonlinear+elasticity">nonlinear elasticity</a></div><div class="keywords"> <h3>Citations:</h3><a href="/0182.478">Zbl 0182.478</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 0689.49006" data-ciurl="/ci/04129491" data-biburl="/bibtex/04129491.bib" data-amsurl="/amsrefs/04129491.bib" data-xmlurl="/xml/04129491.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/04129491.pdf" title="Zbl 0689.49006 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/10.1090/S0273-0979-1989-15818-7" aria-label="DOI for “A surprising higher integrability property of mappings with positive determinant”" title="10.1090/S0273-0979-1989-15818-7">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">John M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal. 63 (1976/77), no. 4, 337 – 403. · <a href="/0368.73040" class="nowrap">Zbl 0368.73040</a> · <a href="https://doi.org/10.1007/BF00279992" class="nowrap">doi:10.1007/BF00279992</a></td> </tr><tr> <td>[2]</td> <td class="space">J. M. Ball and F. Murat, \?^{1,\?}-quasiconvexity and variational problems for multiple integrals, J. Funct. Anal. 58 (1984), no. 3, 225 – 253. , https://doi.org/10.1016/0022-1236(84)90041-7 J. M. Ball and F. Murat, Erratum: ”\?^{1,\?}-quasiconvexity and variational problems for multiple integrals”, J. Funct. Anal. 66 (1986), no. 3, 439. · <a href="https://doi.org/10.1016/0022-1236(86)90068-6" class="nowrap">doi:10.1016/0022-1236(86)90068-6</a></td> </tr><tr> <td>[3]</td> <td class="space">Ronald J. DiPerna and Andrew J. Majda, Oscillations and concentrations in weak solutions of the incompressible fluid equations, Comm. Math. Phys. 108 (1987), no. 4, 667 – 689. · <a href="/0626.35059" class="nowrap">Zbl 0626.35059</a></td> </tr><tr> <td>[4]</td> <td class="space">I. Ekeland and R. Teman, Convex analysis and variational problems, North-Holland, Amsterdam, 1976.</td> </tr><tr> <td>[5]</td> <td class="space">Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. · <a href="/0176.00801" class="nowrap">Zbl 0176.00801</a></td> </tr><tr> <td>[6]</td> <td class="space">P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 2, 109 – 145 (English, with French summary). P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. II, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 4, 223 – 283 (English, with French summary).</td> </tr><tr> <td>[7]</td> <td class="space">P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. I, Rev. Mat. Iberoamericana 1 (1985), no. 1, 145 – 201. · <a href="/0704.49005" class="nowrap">Zbl 0704.49005</a> · <a href="https://doi.org/10.4171/RMI/6" class="nowrap">doi:10.4171/RMI/6</a></td> </tr><tr> <td>[8]</td> <td class="space">S. Müller, Higher integrability of determinants and weak compactness in L1, preprint.</td> </tr><tr> <td>[9]</td> <td class="space">Y. G. Reshetnyak, Stability theorems for mappings with bounded excursion, Siberian Math. J. 9 (1968), 499-512. · <a href="/0176.03503" class="nowrap">Zbl 0176.03503</a></td> </tr><tr> <td>[10]</td> <td class="space">Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. · <a href="/0207.13501" class="nowrap">Zbl 0207.13501</a></td> </tr><tr> <td>[11]</td> <td class="space">E. M. Stein, Note on the class \? \?\?\? \?, Studia Math. 32 (1969), 305 – 310. · <a href="/0182.47803" class="nowrap">Zbl 0182.47803</a></td> </tr><tr> <td>[12]</td> <td class="space">K. W. Zhang, Biting theorems for jacobians and their applications, preprint. · <a href="/0717.49012" class="nowrap">Zbl 0717.49012</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. 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