Strong unique continuation for the higher order fractional Laplacian

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"> <html> <head> <meta name=renderer content=webkit> <meta http-equiv="X-UA-Compatible" content="IE=edge" /> <meta http-equiv="Content-Type" content="text/html; charset=utf-8" /> <meta name="viewport" content="width=device-width,initial-scale=1.0,maximum-scale=1.0,minimum-scale=1.0,user-scalable=no" /> <title>Strong unique continuation for the higher order fractional Laplacian</title> <meta name="_sowise_journalid" content="293118"/> <meta name="WT.cg_n" content="Mathematics in Engineering"/> <meta name="prism.issn" content="2640-3501"/> <meta name="citation_issn" content="2640-3501"/> <meta name="journal_id" content="118"/> <meta name="dc.title" content="Strong unique continuation for the higher order fractional Laplacian"/> <meta name="dc.source" content="Mathematics in Engineering 2019 4:715"/> <meta name="dc.format" content="text/html"/> <meta name="dc.type" content="OriginalPaper"/> <meta name="prism.section" content="OriginalPaper"/> <meta name="prism.publicationName" content="Mathematics in Engineering"/> <meta name="dc.date" content=""/> <meta name="dc.language" content="en"/> <meta name="dc.copyright" content="2019 The Author(s)"/> <meta name="dc.rightsAgent" content="mine@aimspress.org"/> <meta name="prism.rightsAgent" content="mine@aimspress.org"/> <meta name="dc.description" content="In this article we study the strong unique continuation property for solutions of higher order (variable coefficient) fractional Schrödinger operators. We deduce the strong unique continuation property in the presence of subcritical and critical Hardy type potentials. In the same setting, we address the unique continuation property from measurable sets of positive Lebesgue measure. As applications we prove the antilocality of the higher order fractional Laplacian and Runge type approximation theorems which have recently been exploited in the context of nonlocal Calderón type problems. As our main tools, we rely on the characterisation of the higher order fractional Laplacian through a generalised Caffarelli-Silvestre type extension problem and on adapted, iterated Carleman estimates."/> <meta name="og:description" content="In this article we study the strong unique continuation property for solutions of higher order (variable coefficient) fractional Schrödinger operators. We deduce the strong unique continuation property in the presence of subcritical and critical Hardy type potentials. In the same setting, we address the unique continuation property from measurable sets of positive Lebesgue measure. As applications we prove the antilocality of the higher order fractional Laplacian and Runge type approximation theorems which have recently been exploited in the context of nonlocal Calderón type problems. As our main tools, we rely on the characterisation of the higher order fractional Laplacian through a generalised Caffarelli-Silvestre type extension problem and on adapted, iterated Carleman estimates."/> <meta name="description" content="In this article we study the strong unique continuation property for solutions of higher order (variable coefficient) fractional Schrödinger operators. We deduce the strong unique continuation property in the presence of subcritical and critical Hardy type potentials. In the same setting, we address the unique continuation property from measurable sets of positive Lebesgue measure. As applications we prove the antilocality of the higher order fractional Laplacian and Runge type approximation theorems which have recently been exploited in the context of nonlocal Calderón type problems. As our main tools, we rely on the characterisation of the higher order fractional Laplacian through a generalised Caffarelli-Silvestre type extension problem and on adapted, iterated Carleman estimates."/> <meta name="prism.publicationDate" content=""/> <meta name="prism.volume" content="1"/> <meta name="prism.number" content="mine-01-04-715"/> <meta name="prism.startingPage" content="715"/> <meta name="prism.endingPage" content="774"/> <meta name="prism.copyright" content="2019 The Author(s)"/> <meta name="prism.url" content="http://www.aimspress.com/article/doi/10.3934/mine.2019.4.715"/> <meta name="prism.doi" content="doi:10.3934/mine.2019.4.715"/> <meta name="citation_pdf_url" content="http://www.aimspress.com/article/doi/10.3934/mine.2019.4.715"/> <meta name="citation_fulltext_html_url" content="http://www.aimspress.com/article/doi/10.3934/mine.2019.4.715"/> <meta name="citation_journal_title" content="Mathematics in Engineering"/> <meta name="citation_journal_abbrev" content="MINE"/> <meta name="citation_title" content="Strong unique continuation for the higher order fractional Laplacian"/> <meta name="citation_volume" content="1"/> <meta name="citation_issue" content="4"/> <meta name="citation_online_date" content=""/> <meta name="citation_firstpage" content="715"/> <meta name="citation_lastpage" content="774"/> <meta name="citation_article_type" content="Article"/> <meta name="citation_fulltext_world_readable" content=""/> <meta name="citation_language" content="en"/> <meta name="dc.identifier" content="doi:10.3934/mine.2019.4.715"/> <meta name="DOI" content="10.3934/mine.2019.4.715"/> <meta name="citation_doi" content="10.3934/mine.2019.4.715"/> <meta name="dc.subject" content="Research article"/>  <meta name="dc.creator" content="María Ángeles García-Ferrero"/> <meta name="dc.creator" content=" Angkana Rüland"/>  <meta name="citation_reference" content="Chang SYA, Gonzalez MdM (2011) Fractional Laplacian in conformal geometry. <i>Adv Math</i> 226: 1410–1432."/> <meta name="citation_reference" content="Graham CR, Zworski M (2003) Scattering matrix in conformal geometry. <i>Invent Math</i> 152: 89–118."/> <meta name="citation_reference" content="Schild B (1984) A regularity result for polyharmonic variational inequalities with thin obstacles. <i>Ann Scuola Norm-Sci</i> 11: 87–122."/> <meta name="citation_reference" content="Caffarelli LA, Friedman A (1979) The obstacle problem for the biharmonic operator. <i>Ann Scuola Norm-Sci</i> 6: 151–184."/> <meta name="citation_reference" content="Antil H, Khatri R, Warma M (2018) External optimal control of nonlocal PDEs. <i>arXiv preprint arXiv:1811.04515</i>."/> <meta name="citation_reference" content="Biccari U, Hernández-Santamarıa V (2017) Controllability of a one-dimensional fractional heat equation: Theoretical and numerical aspects. <i>hal-01562358v2</i>."/> <meta name="citation_reference" content="Ghosh T, Salo M, Uhlmann G (2016) The Calderón problem for the fractional Schrödinger equation. <i>Anal PDE, in press</i>."/> <meta name="citation_reference" content="Ghosh T, Rüland A, Salo M, et al. (2018) Uniqueness and reconstruction for the fractional Calderón problem with a single measurement. <i>arXiv preprint arXiv:1801.04449</i>."/> <meta name="citation_reference" content="Rüland A, Salo M (2017) The fractional Calderón problem: Low regularity and stability. <i>Nonlinear Anal, in press</i>."/> <meta name="citation_reference" content="Rüland A (2015) Unique continuation for fractional Schrödinger equations with rough potentials. <i>Commun Part Diff Eq</i> 40: 77–114."/> <meta name="citation_reference" content="Yu H (2017) Unique continuation for fractional orders of elliptic equations. <i>Ann PDE</i> 3: 16."/> <meta name="citation_reference" content="Colombini F, Koch H (2010) Strong unique continuation for products of elliptic operators of second order. <i>Trans Am Math Soc</i> 362: 345–355."/> <meta name="citation_reference" content="Koch H, Tataru D (2001) Carleman estimates and unique continuation for second-order elliptic equations with nonsmooth coefficients. <i>Commun Pure Appl Math</i> 54: 339–360."/> <meta name="citation_reference" content="Caffarelli LA, Stinga PR (2016) Fractional elliptic equations, Caccioppoli estimates and regularity. <i>Ann Inst H Poincaré Anal Non Linéaire</i> 33: 767–807."/> <meta name="citation_reference" content="Yang R (2013) On higher order extensions for the fractional Laplacian. <i>arXiv preprint arXiv:1302.4413</i>."/> <meta name="citation_reference" content="Miller K (1974) Nonunique continuation for uniformly parabolic and elliptic equations in self-adjoint divergence form with Hölder continuous coefficients. <i>Arch Ration Mech Anal</i> 54: 105–117."/> <meta name="citation_reference" content="Mandache N (1998) On a counterexample concerning unique continuation for elliptic equations in divergence form. <i>Math Phys Anal Geom</i> 1: 273–292."/> <meta name="citation_reference" content="Seo I (2014) On unique continuation for Schrödinger operators of fractional and higher orders. <i>Math Nachr</i> 5: 699–703."/> <meta name="citation_reference" content="Fall MM, Felli V (2014) Unique continuation property and local asymptotics of solutions to fractional elliptic equations. <i>Commun Part Diff Eq</i> 39: 354–397."/> <meta name="citation_reference" content="Caffarelli L,Silvestre L (2007) An extension problem related to the fractional Laplacian. <i>Commun Partial Diff Eq</i> 32: 1245–1260."/> <meta name="citation_reference" content="Roncal L, Stinga PR (2016) Fractional Laplacian on the torus. <i>Commun Contemp Math</i> 18: 1550033."/> <meta name="citation_reference" content="Felli V, Ferrero A (2018) Unique continuation and classification of blow-up profiles for elliptic systems with Neumann boundary coupling and applications to higher order fractional equations. <i>arXiv preprint arXiv:1810.10765</i>."/> <meta name="citation_reference" content="Felli V, Ferrero A (2018) Unique continuation principles for a higher order fractional Laplace equation. <i>arXiv preprint arXiv:1809.09496</i>."/> <meta name="citation_reference" content="Grubb G (2015) Fractional Laplacians on domains, a development of Hörmander's theory of µ-transmission pseudodifferential operators. <i>Adv Math</i> 268: 478–528."/> <meta name="citation_reference" content="Koch H, Rüland A, Shi W (2016) Higher regularity for the fractional thin obstacle problem. <i>arXiv preprint arXiv:1605.06662</i>."/> <meta name="citation_reference" content="Verch R (1993) Antilocality and a Reeh-Schlieder theorem on manifolds. <i>Lett Math Phys</i> 28: 143–154."/> <meta name="citation_reference" content="Ghosh T, Lin YH, Xiao J (2017) The Calderón problem for variable coefficients nonlocal elliptic operators. <i>Commun Part Diff Eq</i> 42: 1923–1961."/> <meta name="citation_reference" content="Rüland A, Salo M (2017) Quantitative approximation properties for the fractional heat equation. <i>Maths Control Rel Fields, in press</i>."/> <meta name="citation_reference" content="Dipierro S, Savin O, Valdinoci E (2014) All functions are locally s-harmonic up to a small error. <i>arXiv preprint arXiv:1404.3652</i>."/> <meta name="citation_reference" content="Carbotti A, Dipierro S, Valdinoci E (2018) Local density of Caputo-stationary functions of any order. <i>Complex Var Elliptic</i> 1–24."/> <meta name="citation_reference" content="Carbotti A, Dipierro S, Valdinoci E (2018) Local density of solutions of time and space fractional equations. <i>arXiv preprint arXiv:1810.08448</i>."/> <meta name="citation_reference" content="Dipierro S, Savin O, Valdinoci E (2019) Local approximation of arbitrary functions by solutions of nonlocal equations. <i>J Geom Anal</i> 29: 1428–1455."/> <meta name="citation_reference" content="Krylov NV (2018) On the paper "All functions are locally s-harmonic up to a small error" by Dipierro, Savin and Valdinoci. <i>arXiv preprint arXiv:1810.07648</i>."/> <meta name="citation_reference" content="Rüland A (2017) Quantitative invertibility and approximation for the truncated Hilbert and Riesz transforms. <i>Rev Mat Iberoam, in press</i>."/> <meta name="citation_reference" content="Rüland A (2017) On quantitative unique continuation properties of fractional Schrödinger equations: Doubling, vanishing order and nodal domain estimates. <i>Trans Am Math Soc</i> 369: 2311–2362."/> <meta name="citation_reference" content="Koch H, Rüland A, Shi W (2016) The variable coefficient thin obstacle problem: Carleman inequalities. <i>Adv Math</i> 301: 820–866."/> <meta name="citation_reference" content="Rüland A, Wang JN (2018) On the fractional Landis conjecture. <i>J Funct Anal, in press</i>."/> <meta name="citation_reference" content="Adolfsson V, Escauriaza L, Kenig C (1995) Convex domains and unique continuation at the boundary. <i>Rev Mat Iberoam</i> 11: 513–525."/> <meta name="citation_reference" content="Riesz M (1938). Intégrales de Riemann-Liouville et potentiels. Acta Sci Math Szeged 9: 1–42."/> <meta name="citation_reference" content="Isakov V (1990) <i>Inverse Source Problems.</i> American Mathematical Soc."/> <meta name="citation_reference" content="Bellassoued M, Le Rousseau J (2015) Carleman estimates for elliptic operators with complex coefficients. Part I: Boundary value problems. <i>J Math Pures Appl</i> 104: 657–728."/> <meta name="citation_reference" content="Stinga PR, Torrea JL (2010) Extension problem and Harnack's inequality for some fractional operators. <i>Commun Part Diff Eq</i> 35: 2092–2122."/> <meta name="citation_reference" content="Goodman J, Spector D (2018) Some remarks on boundary operators of Bessel extensions. <i>Discrete Cont Dyn-S</i> 11: 493–509."/> <meta name="citation_reference" content="Kwaśnicki M, Mucha J (2018) Extension technique for complete Bernstein functions of the Laplace operator. <i>J Evol Equ</i> 18: 1341–1379."/> <meta name="citation_reference" content="Lin FH, Wang L (1998) A class of fully nonlinear elliptic equations with singularity at the boundary. <i>J Geom Anal</i> 8: 583–598."/> <meta name="citation_reference" content="Abramowitz M,Stegun IA (1965) <i>Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables</i> vol. 55. Courier Corporation."/> <meta name="citation_reference" content="Olver FWJ (2010) <i>NIST Handbook of Mathematical Functions Hardback and CD-ROM.</i> Cambridge University Press."/>  <meta name="citation_author" content="María Ángeles García-Ferrero"/> <meta name="citation_author" content=" Angkana Rüland"/>  <meta name="citation_author_institution" content="Max-Planck-Institute for Mathematics in the Sciences, Inselstr. 22, 04103 Leipzig, Germany"/> <meta name="access_endpoint" content="http://www.aimspress.com/"/> <meta name="twitter:site" content="@Mathematics in Engineering"/> <meta name="twitter:card" content="summary"/> <meta name="twitter:image:alt" content="Content cover image"/> <meta name="WT.z_cc_license_type" content="cc_by"/> <meta name="WT.z_primary_atype" content="Mathematics in Engineering"/> <meta name="WT.z_subject_term" content="Research article"/> <meta name="WT.z_subject_term_id" content="Research article"/> <meta property="og:url" 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class="article-journal fl" > <h3 class="journal-con" ><a href="/journal/mine" class="journal-tit"> Mathematics in Engineering </a></h3> <div class="article-list-time"> <font > 2019, <span><a href="/mine/article/archives" class="mainColor">Volume 1</a>, </span> <a href="/mine/article/2019/4/archive-articles" class="mainColor">Issue 4</a><span>:</span> <span>715-774</span>. </font> <font>doi: <a href="https://doi.org/10.3934/mine.2019.4.715" target="_blank" class="mainColor">10.3934/mine.2019.4.715</a></font> </div> </div> <div class="fr changebtn" > <a href="/article/doi/10.3934/mine.2019.4.699" class="page-btn prev-page fl">Previous Article</a> <a href="/article/doi/10.3934/mine.2019.4.775" class="page-btn next-page fl">Next Article</a> </div> </div> <div class="article-left"> <div class="article-column "> <span class="columnli">Research article</span> <span class="columnli access">Special Issues</span>   </div> <div class="articleEn"> <div class="article-title left-title"> <h1>Strong unique continuation for the higher order fractional Laplacian</h1> </div> <ul class="article-author clear"> <li > <a href="javascript:void(0);" class="mainColor" data-relate="" data="{{author.authorNameEn}}" type="authors.authorNameEn">María Ángeles García-Ferrero</a> <sup class="authorTag"> </sup>,  </li> <li > <a href="javascript:void(0);" class="mainColor" data-relate="rueland@mis.mpg.de" data="{{author.authorNameEn}}" type="authors.authorNameEn"> Angkana Rüland</a> <sup class="authorTag"> <a href="javascript:void(0);" class="com-user" title="Corresponding Author:  Angkana Rüland, rueland@mis.mpg.de"></a>, <a href="mailto:rueland@mis.mpg.de" class="com-mail" title="mailto: rueland@mis.mpg.de"></a></sup> </li> </ul> <ul class="about-author">  <li class="article-author-address"> <span class=""> </span> <div > Max-Planck-Institute for Mathematics in the Sciences, Inselstr. 22, 04103 Leipzig, Germany </div> </li> </ul> <ul class="about-article">  <li class="com-author-info timewrap"> <span><b>Received:</b> 26 February 2019 </span> <span><b>Accepted:</b> 22 June 2019 </span> <span><b>Published:</b> 20 August 2019 </span> </li> <li class="fundwrap"> </li> <li class="com-author-info"> </li> <li class="com-author-info"> <span class="columnli" id="article_special_show" hidden> </span> </li> </p> </li> </ul> </div> </div> <ul id="myTab" class="tab-ul tab-ul-article clear"> <li class="active abs"><a href="#Abstract" data-toggle="tab">Abstract</a></li> <li class="htm"><a href="#FullTextWrap" data-toggle="tab">Full Text(HTML)</a></li>  <li class="pdf hidden-sm hidden-xs"><div class="download-pdf" data="4097"><a href="javascript:void(0);" > Download PDF</a></div></li> <li class="pdf hidden-lg hidden-md"> <a href="/aimspress-data/mine/2019/4/PDF/mine-01-04-715.pdf" > Download PDF </a> </li> </ul> <ul class="article-tab-box tab-content" id="myTabContent">  <li id="Abstract" class="articleListBox tab-pane fade in active">   <div class="article-abstract"> In this article we study the strong unique continuation property for solutions of higher order (variable coefficient) fractional Schrödinger operators. We deduce the strong unique continuation property in the presence of subcritical and critical Hardy type potentials. In the same setting, we address the unique continuation property from measurable sets of positive Lebesgue measure. As applications we prove the antilocality of the higher order fractional Laplacian and Runge type approximation theorems which have recently been exploited in the context of nonlocal Calderón type problems. As our main tools, we rely on the characterisation of the higher order fractional Laplacian through a generalised Caffarelli-Silvestre type extension problem and on adapted, iterated Carleman estimates. </div> <ul class="article-keyword article-info-en"> <b>Keywords:</b> <li><a class="underHigh mainColor" href="javascript:void(0);" data="unique continuation" type="keywords.keywordEn">unique continuation</a>, </li> <li><a class="underHigh mainColor" href="javascript:void(0);" data="fractional Schrödinger equation" type="keywords.keywordEn">fractional Schrödinger equation</a>, </li> <li><a class="underHigh mainColor" href="javascript:void(0);" data="higher order nonlocal operators" type="keywords.keywordEn">higher order nonlocal operators</a>, </li> <li><a class="underHigh mainColor" href="javascript:void(0);" data="Carleman estimates" type="keywords.keywordEn">Carleman estimates</a> </li> </ul> <p class="citation-p"><b class="subtit-b">Citation:</b> María Ángeles García-Ferrero, Angkana Rüland. Strong unique continuation for the higher order fractional Laplacian[J]. Mathematics in Engineering, 2019, 1(4): 715-774. doi: 10.3934/mine.2019.4.715</p>  <div id="RelatedPages" class="articleListBox"> <h3><b class="subtit-b">Related Papers:</b></h3> <div id="RelatedPagesHtml"></div> </div> </li>  <li id="FullTextWrap" class="articleListBox FullText-all tab-pane fade in"> <h3 class="navTitle" id="Abstract-list">Abstract</h3>  <div class="article-abstract"> In this article we study the strong unique continuation property for solutions of higher order (variable coefficient) fractional Schrödinger operators. We deduce the strong unique continuation property in the presence of subcritical and critical Hardy type potentials. In the same setting, we address the unique continuation property from measurable sets of positive Lebesgue measure. As applications we prove the antilocality of the higher order fractional Laplacian and Runge type approximation theorems which have recently been exploited in the context of nonlocal Calderón type problems. As our main tools, we rely on the characterisation of the higher order fractional Laplacian through a generalised Caffarelli-Silvestre type extension problem and on adapted, iterated Carleman estimates. </div> <br /> <div class="nodeTitle"><b></b></div> <br /> <div class="nodeContent"> </div> <div id="FullText" > <img style="display:block;margin:10px auto;" src="/style/web/images/article/1.gif" alt="加载中" /> </div>  </br>   </br> <h3 class="navTitle" id="References-list">References</h3> <div id="References"> <div class="References-wrap"> <table class="reference-tab"> <tr class="document-box" id="b1"> <td valign="top" class="td1"> [1] </td> <td class="td2">  Chang SYA, Gonzalez MdM (2011) Fractional Laplacian in conformal geometry. <i>Adv Math</i> 226: 1410–1432. doi: <a href="https://doi.org/10.1016/j.aim.2010.07.016" target="_blank">10.1016/j.aim.2010.07.016</a> <span><a href="https://doi.org/10.1016/j.aim.2010.07.016" target="_blank" class="crossref-img"><img alt="" src="/style/web/images/custom/crossref.jpeg"></a></span> </td> </tr> <tr class="document-box" id="b2"> <td valign="top" class="td1"> [2] </td> <td class="td2">  Graham CR, Zworski M (2003) Scattering matrix in conformal geometry. <i>Invent Math</i> 152: 89–118. doi: <a href="https://doi.org/10.1007/s00222-002-0268-1" target="_blank">10.1007/s00222-002-0268-1</a> <span><a href="https://doi.org/10.1007/s00222-002-0268-1" target="_blank" class="crossref-img"><img alt="" src="/style/web/images/custom/crossref.jpeg"></a></span> </td> </tr> <tr class="document-box" id="b3"> <td valign="top" class="td1"> [3] </td> <td class="td2">  Schild B (1984) A regularity result for polyharmonic variational inequalities with thin obstacles. <i>Ann Scuola Norm-Sci</i> 11: 87–122. </td> </tr> <tr class="document-box" id="b4"> <td valign="top" class="td1"> [4] </td> <td class="td2">  Caffarelli LA, Friedman A (1979) The obstacle problem for the biharmonic operator. <i>Ann Scuola Norm-Sci</i> 6: 151–184. </td> </tr> <tr class="document-box" id="b5"> <td valign="top" class="td1"> [5] </td> <td class="td2">  Antil H, Khatri R, Warma M (2018) External optimal control of nonlocal PDEs. <i>arXiv preprint arXiv:1811.04515</i>. </td> </tr> <tr class="document-box" id="b6"> <td valign="top" class="td1"> [6] </td> <td class="td2">  Biccari U, Hernández-Santamarıa V (2017) Controllability of a one-dimensional fractional heat equation: Theoretical and numerical aspects. <i>hal-01562358v2</i>. </td> </tr> <tr class="document-box" id="b7"> <td valign="top" class="td1"> [7] </td> <td class="td2">  Ghosh T, Salo M, Uhlmann G (2016) The Calderón problem for the fractional Schrödinger equation. <i>Anal PDE, in press</i>. </td> </tr> <tr class="document-box" id="b8"> <td valign="top" class="td1"> [8] </td> <td class="td2">  Ghosh T, Rüland A, Salo M, et al. (2018) Uniqueness and reconstruction for the fractional Calderón problem with a single measurement. <i>arXiv preprint arXiv:1801.04449</i>. </td> </tr> <tr class="document-box" id="b9"> <td valign="top" class="td1"> [9] </td> <td class="td2">  Rüland A, Salo M (2017) The fractional Calderón problem: Low regularity and stability. <i>Nonlinear Anal, in press</i>. </td> </tr> <tr class="document-box" id="b10"> <td valign="top" class="td1"> [10] </td> <td class="td2">  Rüland A (2015) Unique continuation for fractional Schrödinger equations with rough potentials. <i>Commun Part Diff Eq</i> 40: 77–114. doi: <a href="https://doi.org/10.1080/03605302.2014.905594" target="_blank">10.1080/03605302.2014.905594</a> <span><a href="https://doi.org/10.1080/03605302.2014.905594" target="_blank" class="crossref-img"><img alt="" src="/style/web/images/custom/crossref.jpeg"></a></span> </td> </tr> <tr class="document-box" id="b11"> <td valign="top" class="td1"> [11] </td> <td class="td2">  Yu H (2017) Unique continuation for fractional orders of elliptic equations. <i>Ann PDE</i> 3: 16. doi: <a href="https://doi.org/10.1007/s40818-017-0033-9" target="_blank">10.1007/s40818-017-0033-9</a> <span><a href="https://doi.org/10.1007/s40818-017-0033-9" target="_blank" class="crossref-img"><img alt="" src="/style/web/images/custom/crossref.jpeg"></a></span> </td> </tr> <tr class="document-box" id="b12"> <td valign="top" class="td1"> [12] </td> <td class="td2">  Colombini F, Koch H (2010) Strong unique continuation for products of elliptic operators of second order. <i>Trans Am Math Soc</i> 362: 345–355. </td> </tr> <tr class="document-box" id="b13"> <td valign="top" class="td1"> [13] </td> <td class="td2">  Koch H, Tataru D (2001) Carleman estimates and unique continuation for second-order elliptic equations with nonsmooth coefficients. <i>Commun Pure Appl Math</i> 54: 339–360. doi: <a href="https://doi.org/10.1002/1097-0312(200103)54:3<339::AID-CPA3>3.0.CO;2-D" target="_blank">10.1002/1097-0312(200103)54:3<339::AID-CPA3>3.0.CO;2-D</a> <span><a href="https://doi.org/10.1002/1097-0312(200103)54:3<339::AID-CPA3>3.0.CO;2-D" target="_blank" class="crossref-img"><img alt="" src="/style/web/images/custom/crossref.jpeg"></a></span> </td> </tr> <tr class="document-box" id="b14"> <td valign="top" class="td1"> [14] </td> <td class="td2">  Caffarelli LA, Stinga PR (2016) Fractional elliptic equations, Caccioppoli estimates and regularity. <i>Ann Inst H Poincaré Anal Non Linéaire</i> 33: 767–807. doi: <a href="https://doi.org/10.1016/j.anihpc.2015.01.004" target="_blank">10.1016/j.anihpc.2015.01.004</a> <span><a href="https://doi.org/10.1016/j.anihpc.2015.01.004" target="_blank" class="crossref-img"><img alt="" src="/style/web/images/custom/crossref.jpeg"></a></span> </td> </tr> <tr class="document-box" id="b15"> <td valign="top" class="td1"> [15] </td> <td class="td2">  Yang R (2013) On higher order extensions for the fractional Laplacian. <i>arXiv preprint arXiv:1302.4413</i>. </td> </tr> <tr class="document-box" id="b16"> <td valign="top" class="td1"> [16] </td> <td class="td2">  Miller K (1974) Nonunique continuation for uniformly parabolic and elliptic equations in self-adjoint divergence form with Hölder continuous coefficients. <i>Arch Ration Mech Anal</i> 54: 105–117. doi: <a href="https://doi.org/10.1007/BF00247634" target="_blank">10.1007/BF00247634</a> <span><a href="https://doi.org/10.1007/BF00247634" target="_blank" class="crossref-img"><img alt="" src="/style/web/images/custom/crossref.jpeg"></a></span> </td> </tr> <tr class="document-box" id="b17"> <td valign="top" class="td1"> [17] </td> <td class="td2">  Mandache N (1998) On a counterexample concerning unique continuation for elliptic equations in divergence form. <i>Math Phys Anal Geom</i> 1: 273–292. doi: <a href="https://doi.org/10.1023/A:1009745125885" target="_blank">10.1023/A:1009745125885</a> <span><a href="https://doi.org/10.1023/A:1009745125885" target="_blank" class="crossref-img"><img alt="" src="/style/web/images/custom/crossref.jpeg"></a></span> </td> </tr> <tr class="document-box" id="b18"> <td valign="top" class="td1"> [18] </td> <td class="td2">  Seo I (2014) On unique continuation for Schrödinger operators of fractional and higher orders. <i>Math Nachr</i> 5: 699–703. </td> </tr> <tr class="document-box" id="b19"> <td valign="top" class="td1"> [19] </td> <td class="td2">  Fall MM, Felli V (2014) Unique continuation property and local asymptotics of solutions to fractional elliptic equations. <i>Commun Part Diff Eq</i> 39: 354–397. doi: <a href="https://doi.org/10.1080/03605302.2013.825918" target="_blank">10.1080/03605302.2013.825918</a> <span><a href="https://doi.org/10.1080/03605302.2013.825918" target="_blank" class="crossref-img"><img alt="" src="/style/web/images/custom/crossref.jpeg"></a></span> </td> </tr> <tr class="document-box" id="b20"> <td valign="top" class="td1"> [20] </td> <td class="td2">  Caffarelli L,Silvestre L (2007) An extension problem related to the fractional Laplacian. <i>Commun Partial Diff Eq</i> 32: 1245–1260. doi: <a href="https://doi.org/10.1080/03605300600987306" target="_blank">10.1080/03605300600987306</a> <span><a href="https://doi.org/10.1080/03605300600987306" target="_blank" class="crossref-img"><img alt="" src="/style/web/images/custom/crossref.jpeg"></a></span> </td> </tr> <tr class="document-box" id="b21"> <td valign="top" class="td1"> [21] </td> <td class="td2">  Roncal L, Stinga PR (2016) Fractional Laplacian on the torus. <i>Commun Contemp Math</i> 18: 1550033. doi: <a href="https://doi.org/10.1142/S0219199715500339" target="_blank">10.1142/S0219199715500339</a> <span><a href="https://doi.org/10.1142/S0219199715500339" target="_blank" class="crossref-img"><img alt="" src="/style/web/images/custom/crossref.jpeg"></a></span> </td> </tr> <tr class="document-box" id="b22"> <td valign="top" class="td1"> [22] </td> <td class="td2">  Felli V, Ferrero A (2018) Unique continuation and classification of blow-up profiles for elliptic systems with Neumann boundary coupling and applications to higher order fractional equations. <i>arXiv preprint arXiv:1810.10765</i>. </td> </tr> <tr class="document-box" id="b23"> <td valign="top" class="td1"> [23] </td> <td class="td2">  Felli V, Ferrero A (2018) Unique continuation principles for a higher order fractional Laplace equation. <i>arXiv preprint arXiv:1809.09496</i>. </td> </tr> <tr class="document-box" id="b24"> <td valign="top" class="td1"> [24] </td> <td class="td2">  Grubb G (2015) Fractional Laplacians on domains, a development of Hörmander's theory of µ-transmission pseudodifferential operators. <i>Adv Math</i> 268: 478–528. doi: <a href="https://doi.org/10.1016/j.aim.2014.09.018" target="_blank">10.1016/j.aim.2014.09.018</a> <span><a href="https://doi.org/10.1016/j.aim.2014.09.018" target="_blank" class="crossref-img"><img alt="" src="/style/web/images/custom/crossref.jpeg"></a></span> </td> </tr> <tr class="document-box" id="b25"> <td valign="top" class="td1"> [25] </td> <td class="td2">  Koch H, Rüland A, Shi W (2016) Higher regularity for the fractional thin obstacle problem. <i>arXiv preprint arXiv:1605.06662</i>. </td> </tr> <tr class="document-box" id="b26"> <td valign="top" class="td1"> [26] </td> <td class="td2">  Verch R (1993) Antilocality and a Reeh-Schlieder theorem on manifolds. <i>Lett Math Phys</i> 28: 143–154. doi: <a href="https://doi.org/10.1007/BF00750307" target="_blank">10.1007/BF00750307</a> <span><a href="https://doi.org/10.1007/BF00750307" target="_blank" class="crossref-img"><img alt="" src="/style/web/images/custom/crossref.jpeg"></a></span> </td> </tr> <tr class="document-box" id="b27"> <td valign="top" class="td1"> [27] </td> <td class="td2">  Ghosh T, Lin YH, Xiao J (2017) The Calderón problem for variable coefficients nonlocal elliptic operators. <i>Commun Part Diff Eq</i> 42: 1923–1961. doi: <a href="https://doi.org/10.1080/03605302.2017.1390681" target="_blank">10.1080/03605302.2017.1390681</a> <span><a href="https://doi.org/10.1080/03605302.2017.1390681" target="_blank" class="crossref-img"><img alt="" src="/style/web/images/custom/crossref.jpeg"></a></span> </td> </tr> <tr class="document-box" id="b28"> <td valign="top" class="td1"> [28] </td> <td class="td2">  Rüland A, Salo M (2017) Quantitative approximation properties for the fractional heat equation. <i>Maths Control Rel Fields, in press</i>. </td> </tr> <tr class="document-box" id="b29"> <td valign="top" class="td1"> [29] </td> <td class="td2">  Dipierro S, Savin O, Valdinoci E (2014) All functions are locally s-harmonic up to a small error. <i>arXiv preprint arXiv:1404.3652</i>. </td> </tr> <tr class="document-box" id="b30"> <td valign="top" class="td1"> [30] </td> <td class="td2">  Carbotti A, Dipierro S, Valdinoci E (2018) Local density of Caputo-stationary functions of any order. <i>Complex Var Elliptic</i> 1–24. </td> </tr> <tr class="document-box" id="b31"> <td valign="top" class="td1"> [31] </td> <td class="td2">  Carbotti A, Dipierro S, Valdinoci E (2018) Local density of solutions of time and space fractional equations. <i>arXiv preprint arXiv:1810.08448</i>. </td> </tr> <tr class="document-box" id="b32"> <td valign="top" class="td1"> [32] </td> <td class="td2">  Dipierro S, Savin O, Valdinoci E (2019) Local approximation of arbitrary functions by solutions of nonlocal equations. <i>J Geom Anal</i> 29: 1428–1455. doi: <a href="https://doi.org/10.1007/s12220-018-0045-z" target="_blank">10.1007/s12220-018-0045-z</a> <span><a href="https://doi.org/10.1007/s12220-018-0045-z" target="_blank" class="crossref-img"><img alt="" src="/style/web/images/custom/crossref.jpeg"></a></span> </td> </tr> <tr class="document-box" id="b33"> <td valign="top" class="td1"> [33] </td> <td class="td2">  Krylov NV (2018) On the paper "All functions are locally s-harmonic up to a small error" by Dipierro, Savin and Valdinoci. <i>arXiv preprint arXiv:1810.07648</i>. </td> </tr> <tr class="document-box" id="b34"> <td valign="top" class="td1"> [34] </td> <td class="td2">  Rüland A (2017) Quantitative invertibility and approximation for the truncated Hilbert and Riesz transforms. <i>Rev Mat Iberoam, in press</i>. </td> </tr> <tr class="document-box" id="b35"> <td valign="top" class="td1"> [35] </td> <td class="td2">  Rüland A (2017) On quantitative unique continuation properties of fractional Schrödinger equations: Doubling, vanishing order and nodal domain estimates. <i>Trans Am Math Soc</i> 369: 2311–2362. </td> </tr> <tr class="document-box" id="b36"> <td valign="top" class="td1"> [36] </td> <td class="td2">  Koch H, Rüland A, Shi W (2016) The variable coefficient thin obstacle problem: Carleman inequalities. <i>Adv Math</i> 301: 820–866. doi: <a href="https://doi.org/10.1016/j.aim.2016.06.023" target="_blank">10.1016/j.aim.2016.06.023</a> <span><a href="https://doi.org/10.1016/j.aim.2016.06.023" target="_blank" class="crossref-img"><img alt="" src="/style/web/images/custom/crossref.jpeg"></a></span> </td> </tr> <tr class="document-box" id="b37"> <td valign="top" class="td1"> [37] </td> <td class="td2">  Rüland A, Wang JN (2018) On the fractional Landis conjecture. <i>J Funct Anal, in press</i>. </td> </tr> <tr class="document-box" id="b38"> <td valign="top" class="td1"> [38] </td> <td class="td2">  Adolfsson V, Escauriaza L, Kenig C (1995) Convex domains and unique continuation at the boundary. <i>Rev Mat Iberoam</i> 11: 513–525. </td> </tr> <tr class="document-box" id="b39"> <td valign="top" class="td1"> [39] </td> <td class="td2">  Riesz M (1938). Intégrales de Riemann-Liouville et potentiels. Acta Sci Math Szeged 9: 1–42. </td> </tr> <tr class="document-box" id="b40"> <td valign="top" class="td1"> [40] </td> <td class="td2">  Isakov V (1990) <i>Inverse Source Problems.</i> American Mathematical Soc. </td> </tr> <tr class="document-box" id="b41"> <td valign="top" class="td1"> [41] </td> <td class="td2">  Bellassoued M, Le Rousseau J (2015) Carleman estimates for elliptic operators with complex coefficients. Part I: Boundary value problems. <i>J Math Pures Appl</i> 104: 657–728. </td> </tr> <tr class="document-box" id="b42"> <td valign="top" class="td1"> [42] </td> <td class="td2">  Stinga PR, Torrea JL (2010) Extension problem and Harnack's inequality for some fractional operators. <i>Commun Part Diff Eq</i> 35: 2092–2122. doi: <a href="https://doi.org/10.1080/03605301003735680" target="_blank">10.1080/03605301003735680</a> <span><a href="https://doi.org/10.1080/03605301003735680" target="_blank" class="crossref-img"><img alt="" src="/style/web/images/custom/crossref.jpeg"></a></span> </td> </tr> <tr class="document-box" id="b43"> <td valign="top" class="td1"> [43] </td> <td class="td2">  Goodman J, Spector D (2018) Some remarks on boundary operators of Bessel extensions. <i>Discrete Cont Dyn-S</i> 11: 493–509. </td> </tr> <tr class="document-box" id="b44"> <td valign="top" class="td1"> [44] </td> <td class="td2">  Kwaśnicki M, Mucha J (2018) Extension technique for complete Bernstein functions of the Laplace operator. <i>J Evol Equ</i> 18: 1341–1379. doi: <a href="https://doi.org/10.1007/s00028-018-0444-4" target="_blank">10.1007/s00028-018-0444-4</a> <span><a href="https://doi.org/10.1007/s00028-018-0444-4" target="_blank" class="crossref-img"><img alt="" src="/style/web/images/custom/crossref.jpeg"></a></span> </td> </tr> <tr class="document-box" id="b45"> <td valign="top" class="td1"> [45] </td> <td class="td2">  Lin FH, Wang L (1998) A class of fully nonlinear elliptic equations with singularity at the boundary. <i>J Geom Anal</i> 8: 583–598. doi: <a href="https://doi.org/10.1007/BF02921713" target="_blank">10.1007/BF02921713</a> <span><a href="https://doi.org/10.1007/BF02921713" target="_blank" class="crossref-img"><img alt="" src="/style/web/images/custom/crossref.jpeg"></a></span> </td> </tr> <tr class="document-box" id="b46"> <td valign="top" class="td1"> [46] </td> <td class="td2">  Abramowitz M,Stegun IA (1965) <i>Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables</i> vol. 55. Courier Corporation. </td> </tr> <tr class="document-box" id="b47"> <td valign="top" class="td1"> [47] </td> <td class="td2">  Olver FWJ (2010) <i>NIST Handbook of Mathematical Functions Hardback and CD-ROM.</i> Cambridge University Press. </td> </tr> </table> </div> </div>   </li>  <li id="References" class="articleListBox"> </li>    <li id="Supplements" class="articleListBox tab-pane fade in">  <table class="zgjx-table"> </table> </li>    <li id="citedby-info" class="articleListBox"></li> <li class="commentwrap" > <div class="clearfix new-bio">   </div> <h5 class="comment-title" ><b>Reader Comments</b></h5> <div class="comment-content" > <form id="commentForm" class="comment-form" name="commentform" > <textarea class="comment-textarea" name="textarea" id="textarea"> </textarea> <div class="line-phone common-line clear"> <label class="fl">Your name:<i class="colorRed">*</i></label> <input name="name" id="name" placeholder="" required /> </div> <div class="line-phone common-line clear"> <label class="fl">Email:<i class="colorRed">*</i></label> <input type="email" id="email" name="email" class="fl" required> </div> <input type="submit" class="submit-comment" value="Submit"> </form> </div> </li>  <li> <div id="trendmd-suggestions"></div> <script defer src='//js.trendmd.com/trendmd.min.js' data-trendmdconfig='{"journal_id":"79542","element":"#trendmd-suggestions"}'></script> </li> <li> <div> © 2019 the Author(s), licensee AIMS Press. 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Strong unique continuation for the higher order fractional Laplacian[J]. Mathematics in Engineering, 2019, 1(4): 715-774. doi: 10.3934/mine.2019.4.715</div> <span class="info citationEn" id=""> María Ángeles García-Ferrero, Angkana Rüland. 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Strong unique continuation for the higher order fractional Laplacian