On Landis’ conjecture in the plane when the potential has an exponentially decaying negative part
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- by B. Davey, C. Kenig and J.-N. Wang
- St. Petersburg Math. J. 31 (2020), 337-353
- 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.References
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Bibliographic Information
- B. Davey
- Affiliation: Department of Mathematics, City College of New York, CUNY, New York 10031, New York
- MR Author ID: 1061015
- Email: bdavey@ccny.cuny.edu
- C. Kenig
- Affiliation: Department of Mathematics, University of Chicago, Illinois 60637, Chicago
- MR Author ID: 100230
- Email: cek@math.uchicago.edu
- J.-N. Wang
- Affiliation: Institute of Applied Mathematical Sciences, NCTS, National Taiwan University, Taipei 106, Taiwan
- MR Author ID: 312382
- Email: jnwang@math.ntu.edu.tw
- 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. - © Copyright 2020 American Mathematical Society
- Journal: St. Petersburg Math. J. 31 (2020), 337-353
- MSC (2010): Primary 35B60, 35J10
- DOI: https://doi.org/10.1090/spmj/1600
- MathSciNet review: 3937504
Dedicated: Dedicated to Vladimir Maz′ya on the occasion of his $80$th birthday