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On quantitative unique continuation properties of fractional Schrödinger equations: Doubling, vanishing order and nodal domain estimates

Author: Angkana Rüland
Journal: Trans. Amer. Math. Soc. 369 (2017), 2311-2362
MSC (2010): Primary 35R11, 35A02
Published electronically: June 20, 2016
MathSciNet review: 3592513
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Abstract: In this article we determine bounds on the maximal order of vanishing for eigenfunctions of a generalized Dirichlet-to-Neumann map (which is associated with fractional Schrödinger equations) on a compact, smooth Riemannian manifold, $ (M,g)$, without boundary. Moreover, with only slight modifications these results generalize to equations with $ C^1$ potentials. Here Carleman estimates are a key tool. These yield a quantitative three balls inequality which implies quantitative bulk and boundary doubling estimates and hence leads to the control of the maximal order of vanishing. Using the boundary doubling property, we prove upper bounds on the $ \mathcal {H}^{n-1}$-measure of nodal domains of eigenfunctions of the generalized Dirichlet-to-Neumann map on analytic manifolds.

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  • [Bak12] Laurent Bakri, Quantitative uniqueness for Schrödinger operator, Indiana Univ. Math. J. 61 (2012), no. 4, 1565-1580. MR 3085618,
  • [BL15] Katarína Bellová and Fang-Hua Lin, Nodal sets of Steklov eigenfunctions, Calc. Var. Partial Differential Equations 54 (2015), no. 2, 2239-2268. MR 3396451,
  • [Brü78] Jochen Brüning, Über Knoten von Eigenfunktionen des Laplace-Beltrami-Operators, Math. Z. 158 (1978), no. 1, 15-21 (German). MR 0478247
  • [CK10] Ferruccio Colombini and Herbert Koch, Strong unique continuation for products of elliptic operators of second order, Trans. Amer. Math. Soc. 362 (2010), no. 1, 345-355. MR 2550154 (2011c:35073),
  • [CM11] Tobias H. Colding and William P. Minicozzi II, Lower bounds for nodal sets of eigenfunctions, Comm. Math. Phys. 306 (2011), no. 3, 777-784. MR 2825508,
  • [CS07] Luis Caffarelli and Luis Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007), no. 7-9, 1245-1260. MR 2354493 (2009k:35096),
  • [Dav14] Blair Davey, Some quantitative unique continuation results for eigenfunctions of the magnetic Schrödinger operator, Comm. Partial Differential Equations 39 (2014), no. 5, 876-945. MR 3196190,
  • [DF88] Harold Donnelly and Charles Fefferman, Nodal sets of eigenfunctions on Riemannian manifolds, Invent. Math. 93 (1988), no. 1, 161-183. MR 943927 (89m:58207),
  • [EA97] Vilhelm Adolfsson and Luis Escauriaza, $ C^{1,\alpha }$ domains and unique continuation at the boundary, Comm. Pure Appl. Math. 50 (1997), no. 10, 935-969. MR 1466583 (98m:31003),$ \langle $935::AID-CPA1$ \rangle $3.0.CO;2-H
  • [FF13] Mouhamed Moustapha Fall and Veronica Felli, Unique continuation property and local asymptotics of solutions to fractional elliptic equations, Comm. Partial Differential Equations 39 (2014), no. 2, 354-397. MR 3169789,
  • [GL87] Nicola Garofalo and Fang-Hua Lin, Unique continuation for elliptic operators: a geometric-variational approach, Comm. Pure Appl. Math. 40 (1987), no. 3, 347-366. MR 882069 (88j:35046),
  • [GT01] David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. MR 1814364 (2001k:35004)
  • [Her77] Ira W. Herbst, Spectral theory of the operator $ (p^{2}+m^{2})^{1/2}-Ze^{2}/r$, Comm. Math. Phys. 53 (1977), no. 3, 285-294. MR 0436854 (55 #9790)
  • [HL10] Qing Han and Fang-Hua Lin, Nodal sets of solutions of elliptic differential equations, Books available on Han's homepage, 2010.
  • [HS89] Robert Hardt and Leon Simon, Nodal sets for solutions of elliptic equations, J. Differential Geom. 30 (1989), no. 2, 505-522. MR 1010169 (90m:58031)
  • [HS11] Hamid Hezari and Christopher D. Sogge, A natural lower bound for the size of nodal sets, Anal. PDE 5 (2012), no. 5, 1133-1137. MR 3022851,
  • [HW12] Hamid Hezari and Zuoqin Wang, Lower bounds for volumes of nodal sets: an improvement of a result of Sogge-Zelditch, Spectral geometry, Proc. Sympos. Pure Math., vol. 84, Amer. Math. Soc., Providence, RI, 2012, pp. 229-235. MR 2985319,
  • [JK85] David Jerison and Carlos E. Kenig, Unique continuation and absence of positive eigenvalues for Schrödinger operators, Ann. of Math. (2) 121 (1985), no. 3, 463-494. With an appendix by E. M. Stein. MR 794370 (87a:35058),
  • [K$^+$98] Igor Kukavica, Quantitative uniqueness for second-order elliptic operators, Duke Math. J. 91 (1998), no. 2, 225-240. MR 1600578 (99a:35046),
  • [Kat96] Keiichi Kato, New idea for proof of analyticity of solutions to analytic nonlinear elliptic equations, SUT J. Math. 32 (1996), no. 2, 157-161. MR 1431263 (97k:35064)
  • [KEA95] Vilhelm Adolfsson, Luis Escauriaza, and Carlos Kenig, Convex domains and unique continuation at the boundary, Rev. Mat. Iberoamericana 11 (1995), no. 3, 513-525. MR 1363203 (96j:31003),
  • [KT01] Herbert Koch and Daniel Tataru, Carleman estimates and unique continuation for second-order elliptic equations with nonsmooth coefficients, Comm. Pure Appl. Math. 54 (2001), no. 3, 339-360. MR 1809741 (2001m:35075),$ \langle $339::AID-CPA3$ \rangle $3.0.CO;2-D
  • [KW98] Carlos E. Kenig and Wensheng Wang, A note on boundary unique continuation for harmonic functions in non-smooth domains, Potential Anal. 8 (1998), no. 2, 143-147. MR 1618438 (99c:35048),
  • [Lin91] Fang-Hua Lin, Nodal sets of solutions of elliptic and parabolic equations, Comm. Pure Appl. Math. 44 (1991), no. 3, 287-308. MR 1090434 (92b:58224),
  • [LM73] J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. III, Springer-Verlag, New York-Heidelberg, 1973. Translated from the French by P. Kenneth; Die Grundlehren der mathematischen Wissenschaften, Band 183. MR 0350179
  • [LR95] G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur, Comm. Partial Differential Equations 20 (1995), no. 1-2, 335-356 (French). MR 1312710 (95m:93045),
  • [LZ98] Gilles Lebeau and Enrique Zuazua, Null-controllability of a system of linear thermoelasticity, Arch. Rational Mech. Anal. 141 (1998), no. 4, 297-329. MR 1620510 (99f:93013),
  • [OLBC10] Frank Olver, Daniel Lozier, Ronald Boisvert, and Charles Clark (eds.), NIST handbook of mathematical functions, U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge, 2010. With 1 CD-ROM (Windows, Macintosh and UNIX). MR 2723248 (2012a:33001)
  • [Rül14a] Angkana Rüland. On Some Rigidity Properties in PDEs, Dissertation, University of Bonn, 2014.
  • [Rül14b] Angkana Rüland, Unique continuation for fractional Schrödinger equations with rough potentials, Comm. Partial Differential Equations 40 (2015), no. 1, 77-114. MR 3268922,
  • [Seo14] Ihyeok Seo, On unique continuation for Schrödinger operators of fractional and higher orders, Math. Nachr. 287 (2014), no. 5-6, 699-703. MR 3193945,
  • [Seo15] Ihyeok Seo, Unique continuation for fractional Schrödinger operators in three and higher dimensions, Proc. Amer. Math. Soc. 143 (2015), no. 4, 1661-1664. MR 3314078,
  • [Ste13] Stefan Steinerberger, Lower bounds on nodal sets of eigenfunctions via the heat flow, Comm. Partial Differential Equations 39 (2014), no. 12, 2240-2261. MR 3259555,
  • [SZ11] Christopher D. Sogge and Steve Zelditch, Lower bounds on the Hausdorff measure of nodal sets, Math. Res. Lett. 18 (2011), no. 1, 25-37. MR 2770580 (2012c:58055),
  • [Yau82] Shing-Tung Yau (ed.), Seminar on Differential Geometry, Annals of Mathematics Studies, vol. 102, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1982. Papers presented at seminars held during the academic year 1979-1980. MR 645728 (83a:53002)
  • [Zel14] Steve Zelditch, Measure of nodal sets of analytic Steklov eigenfunctions, arXiv preprint arXiv:1403.0647, 2014.
  • [Zhu13] Jiuyi Zhu, Quantitative uniqueness of elliptic equations, arXiv preprint arXiv:1312.0576, 2013.
  • [Zhu15] Jiuyi Zhu, Doubling property and vanishing order of Steklov eigenfunctions, Comm. Partial Differential Equations 40 (2015), no. 8, 1498-1520. MR 3355501,

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Additional Information

Angkana Rüland
Affiliation: Mathematical Institute of the University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, OX2 6GG Oxford, United Kingdom

Keywords: Quantitative unique continuation, fractional Schr\"odinger equations, Carleman estimates, eigenfunctions, nodal domains
Received by editor(s): July 6, 2014
Received by editor(s) in revised form: March 28, 2015
Published electronically: June 20, 2016
Article copyright: © Copyright 2016 American Mathematical Society

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