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St. Petersburg Mathematical Journal

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On Landis' conjecture in the plane when the potential has an exponentially decaying negative part


Authors: B. Davey, C. Kenig and J.-N. Wang
Original publication: Algebra i Analiz, tom 31 (2019), nomer 2.
Journal: St. Petersburg Math. J. 31 (2020), 337-353
MSC (2010): Primary 35B60, 35J10
DOI: https://doi.org/10.1090/spmj/1600
Published electronically: February 4, 2020
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Abstract: In this article, we continue our investigation into the unique continuation properties of real-valued solutions to elliptic equations in the plane. More precisely, we make another step towards proving a quantitative version of Landis' conjecture by establishing unique continuation at infinity estimates for solutions to equations of the form $ - \Delta u + V u = 0$ in $ \mathbb{R}^2$, where $ V = V_+ - V_-$, $ V_+ \in L^\infty $, and $ V_-$ is a nontrivial function that exhibits exponential decay at infinity. The main tool in the proof of this theorem is an order of vanishing estimate in combination with an iteration scheme. To prove the order of vanishing estimate, we establish a similarity principle for vector-valued Beltrami systems.


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Additional Information

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

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

J.-N. 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/spmj/1600
Keywords: Landis' conjecture, quantitative unique continuation, order of vanishing, vector-valued Beltrami system
Received by editor(s): September 6, 2018
Published electronically: February 4, 2020
Additional Notes: The first author was partially supported by the Simons Foundation Grant 430198.
The second author was partially supported by NSF DMS-1265249.
The third author was partially supported by MOST 105-2115-M-002-014-MY3.
Dedicated: Dedicated to Vladimir Maz′ya on the occasion of his $80$th birthday
Article copyright: © Copyright 2020 American Mathematical Society