Document Zbl 07843763

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Am. Math. Soc.</a> <a href="/?q=in%3A511898" title="Articles in this Issue">152, No. 6, 2529-2539 (2024)</a>. </div> <div class="abstract">Let $N\ge2$ be a natural number and $s\in(0,1)$. The authors consider the fractional relativistic operator $(-\Delta +1)^s$ defined as \[ (-\Delta +1)^s u(x) = C_{N,s}\text{ P.V.}\int_{\mathbb{R}^N} \frac{u(x)-u(y)}{|x-y|^{N/2+s}} K_{N/2+s}(|x-y|)\,dy+u(x) \] for $x\in\mathbb{R}^N$, where P.V. stands for for the Cauchy principal value, $C_{N,s}=2^{-N/2-s+1}\pi^{-N/2}2^{2s}\frac{s(1-s)}{\Gamma(2-s)}$, $\Gamma$ is the Gamma function, and $K_\nu$ denotes the modified Bessel function of the third kind of order $\nu\in\mathbb{R}$ which can be represented by $K_\nu(r)=2^{-\nu-1}r^\nu\int_0^\infty \text{e}^{-t-\frac{r^2}{4t}} t^{-\nu-1}\,dt$. <br class="zbmathjax-paragraph">When $-\Delta+1$ is replaced by $-\Delta$, then $(-\Delta+1)^s$ boils down to the fractional Laplacian operator $(-\Delta)^s$ and $(-\Delta)^s u\to -\Delta u$ as $s\to 1{^-}$ for all $u\in C_0^\infty(\mathbb{R}^N)$. Analogously, $(-\Delta+1)^s$ converges pointwise to $-\Delta+1$ as $s\to 1{^-}$ (see [<span class="zbmathjax-textit">E. Di Nezza</span> et al., Bull Sci. Math. 136, No. 5, 521–573 (2012; <a href="/1252.46023">Zbl 1252.46023</a>]). <br class="zbmathjax-paragraph">The authors study the limiting behaviour of the fractional relativistic seminorm \[ \frac12 C_{N,s} \omega_N\iint_{\Omega\times\Omega}\frac{u(x)-u(y)}{|x-y|^{N/2+s}} K_{N/2+s}(|x-y|)\,dx\,dy \] as $s\to -1^-$ where $\Omega\subset\mathbb{R}^N$ is a bounded domain with a Lipschitz boundary and $\omega_N$ is the volume of the unit ball in $\mathbb{R}^N$. Inspired by [<span class="zbmathjax-textit">J. Bourgain</span> et. al., in: Optimal control and partial differential equations. Amsterdam: IOS Press, 439–455 (2001; <a href="/1103.46310">Zbl 1103.46310</a>)] they prove that for every $u \in H^1(\Omega)$ the seminorm tends to $\mathcal{K}_N\int_\Omega |\nabla u|^2\,dx$ with $\mathcal{K}_N=\frac12\int_{\mathbb{S}^{N-1}} |\omega\cdot \mathbf{e}|^2 \,d\mathcal{H}^{N-1}(\omega)$ where $\mathbb{S}^{N-1}$ is the unit sphere in $\mathbb{R}^N$, $\mathcal{H}^{N-1}$ denotes the $(N-1)$-dimensional Hausdorff measure, $\mathbf{e}$ is any unit vector in $\mathbb{R}^N$ and $\cdot$ stands for the scalar product in $\mathbb{R}^N$. An analogous result is obtained for $u\in H_0^1(\Omega)$.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A2959">Jiří Rákosník (Praha)</a></div> <div class="clearfix"></div></div> <div class="clear"></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%3A33C10" title="MSC2020">33C10</a> </td> <td class="space"> Bessel and Airy functions, cylinder functions, ${}_0F_1$ </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Afractional+relativistic+operator">fractional relativistic operator</a>; <a href="/?q=ut%3ABessel+functions">Bessel functions</a>; <a href="/?q=ut%3Aasymptotic+behaviour">asymptotic behaviour</a></div><div class="keywords"> <h3>Citations:</h3><a href="/1252.46023">Zbl 1252.46023</a>; <a href="/1103.46310">Zbl 1103.46310</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 07843763" data-ciurl="/ci/07843763" data-biburl="/bibtex/07843763.bib" data-amsurl="/amsrefs/07843763.bib" data-xmlurl="/xml/07843763.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07843763.pdf" title="Zbl 07843763 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/proc/16734" aria-label="DOI for “A Bourgain-Brezis-Mironescu type result for the fractional relativistic seminorm”" title="10.1090/proc/16734">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">Ambrosio, Vincenzo, On some convergence results for fractional periodic Sobolev spaces, Opuscula Math., 5-20, 2020 · <a href="/1450.46016" class="nowrap">Zbl 1450.46016</a> · <a href="https://doi.org/10.7494/opmath.2020.40.1.5" class="nowrap">doi:10.7494/opmath.2020.40.1.5</a></td> </tr><tr> <td>[2]</td> <td class="space">Ambrosio, Vincenzo, Nonlinear fractional Schr\"{o}dinger equations in $\mathbb{R}^N$, Frontiers in Elliptic and Parabolic Problems, xvii+662 pp., [2021] ©2021, Birkh\"{a}user/Springer, Cham · <a href="/1472.35003" class="nowrap">Zbl 1472.35003</a> · <a href="https://doi.org/10.1007/978-3-030-60220-8" class="nowrap">doi:10.1007/978-3-030-60220-8</a></td> </tr><tr> <td>[3]</td> <td class="space">Ambrosio, Vincenzo, On the fractional relativistic Schr\"{o}dinger operator, J. 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