Document Zbl 1358.46033

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(Formule de Brézis-Bourgain-Mironescu pour des opérateurs magnétiques.)</strong> <i>(English. French summary)</i> <a class="label nowrap" href="/1358.46033">Zbl 1358.46033</a> </h2> <div class="source"> <a href="/serials/2925" title="Journal Profile">C. R., Math., Acad. Sci. Paris</a> <a href="/?q=in%3A357135" title="Articles in this Issue">354, No. 8, 825-831 (2016)</a>. </div> <div class="abstract"><div class="pre">Let $A:\mathbb{R}^N\to\mathbb{R}^N$ be $C^2$, and let $\Omega\subset\mathbb{R}^N$ be a bounded domain with Lipschitz boundary. The main theorem of the present paper says that \[ \begin{aligned} & \lim_{s\nearrow 1}(1-s)\int_\Omega\int_\Omega \frac{|u(x)-e^{i(x-y)A({\frac{x+y}{2}})}u(y)|^2}{|x-y|^{N+2s}}\,dx\, dy\\ & =K_N\int_\Omega|\nabla u-iA(x)u|^2\, dx\end{aligned} \] for all $u\in H^1_A(\Omega)$. Here, $K_N>0$ is an explicitly given constant, and the space $H^1_A(\Omega)$ consists of all $u\in L^2(\Omega)$ such that the integral on the right is finite. This is a “magnetic” version of the result from <span class="zbmathjax-textit">J. Bourgain</span> et al. [J. Anal. Math. 87, 77–101 (2002; <a href="/1029.46030">Zbl 1029.46030</a>)].</div><div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A7188">Thomas J. Bartsch (Gießen)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A6606062">Cited in <strong>2</strong> Reviews</a></div><div class="clear"><a href="/?q=rf%3A6606062">Cited in <strong>61</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></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Afractional+magnetic+operator">fractional magnetic operator</a>; <a href="/?q=ut%3Afractional+Sobolev+space">fractional Sobolev space</a>; <a href="/?q=ut%3Alimiting+embedding+theorem">limiting embedding theorem</a></div><div class="keywords"> <h3>Citations:</h3><a href="/1029.46030">Zbl 1029.46030</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 1358.46033" data-ciurl="/ci/06606062" data-biburl="/bibtex/06606062.bib" data-amsurl="/amsrefs/06606062.bib" data-xmlurl="/xml/06606062.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06606062.pdf" title="Zbl 1358.46033 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/10.1016/j.crma.2016.04.013" aria-label="DOI for “Bourgain-Brézis-Mironescu formula for magnetic operators”" title="10.1016/j.crma.2016.04.013">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/1602.02967" title="Note: arXiv document may differ from published version">arXiv</a> <a class="btn btn-default btn-xs" type="button" href="http://www.numdam.org/item?id=10.1016/j.crma.2016.04.013" title="Open Access at Numdam">Numdam</a> </div> <div class="sfx" style="float: right;"> <img tabindex="0" src="/static/img/unlock_up.svg" class="cc-license-icon" title="Open version available"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Arioli, G.; Szulkin, A., A semilinear Schrödinger equation in the presence of a magnetic field, Arch. Ration. Mech. Anal., 170, 277-295 (2003) · <a href="/1051.35082" class="nowrap">Zbl 1051.35082</a></td> </tr><tr> <td>[2]</td> <td class="space">Avron, J.; Herbst, I.; Simon, B., Schrödinger operators with magnetic fields. I. General interactions, Duke Math. J., 45, 847-883 (1978) · <a href="/0399.35029" class="nowrap">Zbl 0399.35029</a></td> </tr><tr> <td>[3]</td> <td class="space">Bourgain, J.; Brezis, H.; Mironescu, P., Another look at Sobolev spaces, (Menaldi, J. L.; Rofman, E.; Sulem, A., Optimal Control and Partial Differential Equations. A Volume in Honor of Professor Alain Bensoussan’s 60th Birthday (2001), IOS Press: IOS Press Amsterdam), 439-455 · <a href="/1103.46310" class="nowrap">Zbl 1103.46310</a></td> </tr><tr> <td>[4]</td> <td class="space">Bourgain, J.; Brezis, H.; Mironescu, P., Limiting embedding theorems for $W^{s, p}$ when $s \uparrow 1$ and applications, J. Anal. Math., 87, 77-101 (2002) · <a href="/1029.46030" class="nowrap">Zbl 1029.46030</a></td> </tr><tr> <td>[5]</td> <td class="space">Brasco, L.; Parini, E.; Squassina, M., Stability of variational eigenvalues for the fractional $p$-Laplacian, Discrete Contin. Dyn. Syst., Ser. A, 36, 1813-1845 (2016) · <a href="/1336.35270" class="nowrap">Zbl 1336.35270</a></td> </tr><tr> <td>[6]</td> <td class="space">d’Avenia, P.; Squassina, M., Ground states for fractional magnetic operators · <a href="/1400.49059" class="nowrap">Zbl 1400.49059</a></td> </tr><tr> <td>[7]</td> <td class="space">Evans, L. C., Partial Differential Equations, Graduate Series in Mathematics (2010), American Mathematical Society · <a href="/1194.35001" class="nowrap">Zbl 1194.35001</a></td> </tr><tr> <td>[8]</td> <td class="space">Frank, R. L.; Lieb, E. H.; Seiringer, R., Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators, J. Amer. Math. Soc., 21, 925-950 (2008) · <a href="/1202.35146" class="nowrap">Zbl 1202.35146</a></td> </tr><tr> <td>[9]</td> <td class="space">Ichinose, T., Magnetic relativistic Schrödinger operators and imaginary-time path integrals, (Mathematical Physics, Spectral Theory and Stochastic Analysis. Mathematical Physics, Spectral Theory and Stochastic Analysis, Operator Theory: Advances and Applications, vol. 232 (2013), Birkhäuser/Springer: Birkhäuser/Springer Basel, Switzerland), 247-297 · <a href="/1266.81079" class="nowrap">Zbl 1266.81079</a></td> </tr><tr> <td>[10]</td> <td class="space">Maz’ya, V.; Shaposhnikova, T., On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces, J. Funct. Anal., 195, 230-238 (2002) · <a href="/1028.46050" class="nowrap">Zbl 1028.46050</a></td> </tr><tr> <td>[11]</td> <td class="space">Reed, M.; Simon, B., Methods of Modern Mathematical Physics, I, Functional Analysis (1980), Academic Press, Inc.: Academic Press, Inc. New York · <a href="/0459.46001" class="nowrap">Zbl 0459.46001</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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