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The Landis conjecture for variable coefficient second-order elliptic PDEs


Authors: Blair Davey, Carlos Kenig and Jenn-Nan Wang
Journal: Trans. Amer. Math. Soc. 369 (2017), 8209-8237
MSC (2010): Primary 35B60, 35J10
DOI: https://doi.org/10.1090/tran/7073
Published electronically: July 7, 2017
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Abstract:

In this article, we study the quantitative form of Landis' conjecture in the plane for second-order elliptic equations with variable coefficients. Precisely, let $ A$ be a symmetric, positive-definite matrix with Lipschitz coefficients. Assume that $ V\ge 0$ is a measurable, real-valued function satisfying $ \left \vert \left \vert V\right \vert \right \vert _{L^\infty (\mathbb{R}^2)} \le 1$. Let $ u$ be a real-valued solution to $ \operatorname {div} (A \nabla u) - V u = 0$ in $ \mathbb{R}^2$. If $ u$ is bounded and normalized in the sense that $ \left \vert u(z)\right \vert \le \exp (c_0 \vert z\vert)$ and $ u(0) = 1$, then for any $ R$ sufficiently large,

$\displaystyle \inf _{\vert z_0\vert = R} \Vert u\Vert _{L^\infty (B_1(z_0))} \ge \exp (- C R \log R). $

In addition to equations with electric potentials, we also derive similar estimates for equations with first-order terms, or magnetic potentials. The proofs rely on transforming the equations to Beltrami systems and applying a generalization of Hadamard's three-circle theorem.

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  • [1] Giovanni Alessandrini and Luis Escauriaza, Null-controllability of one-dimensional parabolic equations, ESAIM Control Optim. Calc. Var. 14 (2008), no. 2, 284-293. MR 2394511, https://doi.org/10.1051/cocv:2007055
  • [2] W. O. Amrein, A.-M. Berthier, and V. Georgescu, $ L^{p}$-inequalities for the Laplacian and unique continuation, Ann. Inst. Fourier (Grenoble) 31 (1981), no. 3, vii, 153-168 (English, with French summary). MR 638622
  • [3] B. V. Bojarski, , , , , , and Generalized solutions of a system of differential equations of the first order and elliptic type with discontinuous coefficients, Report. University of Jyväskylä Department of Mathematics and Statistics, vol. 118, University of Jyväskylä, Jyväskylä, 2009. Translated from the 1957 Russian original; With a foreword by Eero Saksman. MR 2488720
  • [4] Jean Bourgain and Carlos E. Kenig, On localization in the continuous Anderson-Bernoulli model in higher dimension, Invent. Math. 161 (2005), no. 2, 389-426. MR 2180453, https://doi.org/10.1007/s00222-004-0435-7
  • [5] 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, https://doi.org/10.1080/03605302.2013.796380
  • [6] Blair Davey, A Meshkov-type construction for the borderline case, Differential Integral Equations 28 (2015), no. 3-4, 271-290. MR 3306563
  • [7] Harold Donnelly and Charles Fefferman, Nodal sets of eigenfunctions on Riemannian manifolds, Invent. Math. 93 (1988), no. 1, 161-183. MR 943927, https://doi.org/10.1007/BF01393691
  • [8] Nicola Garofalo and Fang-Hua Lin, Monotonicity properties of variational integrals, $ A_p$ weights and unique continuation, Indiana Univ. Math. J. 35 (1986), no. 2, 245-268. MR 833393, https://doi.org/10.1512/iumj.1986.35.35015
  • [9] 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, https://doi.org/10.1002/cpa.3160400305
  • [10] Mariano Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies, vol. 105, Princeton University Press, Princeton, NJ, 1983. MR 717034
  • [11] 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
  • [12] David Jerison, Carlos E. Kenig, and 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, https://doi.org/10.2307/1971205
  • [13] Carlos Kenig, Luis Silvestre, and Jenn-Nan Wang, On Landis' conjecture in the plane, Comm. Partial Differential Equations 40 (2015), no. 4, 766-789. MR 3299355, https://doi.org/10.1080/03605302.2014.978015
  • [14] Carlos Kenig and Jenn-Nan Wang, Quantitative uniqueness estimates for second order elliptic equations with unbounded drift, Math. Res. Lett. 22 (2015), no. 4, 1159-1175. MR 3391881, https://doi.org/10.4310/MRL.2015.v22.n4.a10
  • [15] Carlos E. Kenig and Wei-Ming Ni, On the elliptic equation $ Lu-k+K\,{\rm exp}[2u]=0$, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 12 (1985), no. 2, 191-224. MR 829052
  • [16] V. A. Kondrat'ev and E. M. Landis, Qualitative properties of the solutions of a second-order nonlinear equation, Mat. Sb. (N.S.) 135(177) (1988), no. 3, 346-360, 415.
  • [17] Igor Kukavica, Quantitative uniqueness for second-order elliptic operators, Duke Math. J. 91 (1998), no. 2, 225-240. MR 1600578, https://doi.org/10.1215/S0012-7094-98-09111-6
  • [18] Ching-Lung Lin and Jenn-Nan Wang, Quantitative uniqueness estimates for the general second order elliptic equations, J. Funct. Anal. 266 (2014), no. 8, 5108-5125. MR 3177332, https://doi.org/10.1016/j.jfa.2014.02.016
  • [19] V. Z. Meshkov, On the possible rate of decrease at infinity of the solutions of second-order partial differential equations, Mat. Sb. 182 (1991), no. 3, 364-383 (Russian); English transl., Math. USSR-Sb. 72 (1992), no. 2, 343-361. MR 1110071
  • [20] M. Schechter and B. Simon, Unique continuation for Schrödinger operators with unbounded potentials, J. Math. Anal. Appl. 77 (1980), no. 2, 482-492. MR 593229, https://doi.org/10.1016/0022-247X(80)90242-5

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Additional Information

Blair Davey
Affiliation: Department of Mathematics, City College of New York CUNY, New York, New York 10031
Email: bdavey@ccny.cuny.edu

Carlos Kenig
Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois 60637
Email: cek@math.uchicago.edu

Jenn-Nan Wang
Affiliation: Institute of Applied Mathematical Sciences, NCTS, National Taiwan University, Taipei 106, Taiwan
Email: jnwang@math.ntu.edu.tw

DOI: https://doi.org/10.1090/tran/7073
Keywords: Landis' conjecture, Beltrami system, maximal vanishing order
Received by editor(s): October 22, 2015
Received by editor(s) in revised form: April 30, 2016
Published electronically: July 7, 2017
Additional Notes: The second author was supported in part by DMS-1265429.
The third author was supported in part by MOST 102-2115-M-002-009-MY3.
Article copyright: © Copyright 2017 American Mathematical Society

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