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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
Published electronically: July 7, 2017
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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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Additional Information

Blair Davey
Affiliation: Department of Mathematics, City College of New York CUNY, New York, New York 10031

Carlos Kenig
Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois 60637

Jenn-Nan Wang
Affiliation: Institute of Applied Mathematical Sciences, NCTS, National Taiwan University, Taipei 106, Taiwan

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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