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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Certain Liouville properties of eigenfunctions of elliptic operators
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by Ari Arapostathis, Anup Biswas and Debdip Ganguly PDF
Trans. Amer. Math. Soc. 371 (2019), 4377-4409 Request permission

Abstract:

We present certain Liouville properties of eigenfunctions of second-order elliptic operators with real coefficients, via an approach that is based on stochastic representations of positive solutions, and criticality theory of second-order elliptic operators. These extend results of Y. Pinchover to the case of nonsymmetric operators of Schrödinger type. In particular, we provide an answer to an open problem posed by Pinchover in [Comm. Math. Phys. 272 (2007), pp. 75–84, Problem 5]. In addition, we prove a lower bound on the decay of positive supersolutions of general second-order elliptic operators in any dimension, and discuss its implications to the Landis conjecture.
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Additional Information
  • Ari Arapostathis
  • Affiliation: Department of Electrical and Computer Engineering, The University of Texas at Austin, 2501 Speedway, EER 7.824, Austin, Texas 78712
  • MR Author ID: 26760
  • Email: ari@ece.utexas.edu
  • Anup Biswas
  • Affiliation: Department of Mathematics, Indian Institute of Science Education and Research, Dr. Homi Bhabha Road, Pune 411008, India
  • MR Author ID: 866719
  • Email: anup@iiserpune.ac.in
  • Debdip Ganguly
  • Affiliation: Department of Mathematics, Indian Institute of Science Education and Research, Dr. Homi Bhabha Road, Pashan, Pune 411008, India
  • MR Author ID: 1060705
  • Email: debdip@iiserpune.ac.in
  • Received by editor(s): November 16, 2017
  • Received by editor(s) in revised form: June 9, 2018, and June 25, 2018
  • Published electronically: November 26, 2018
  • Additional Notes: The research of the first author was supported in part by the Army Research Office through grant W911NF-17-1-001, in part by the National Science Foundation through grant DMS-1715210, and in part by Office of Naval Research through grant N00014-16-1-2956.
    The research of the second author was supported in part by an INSPIRE faculty fellowship (IFA13/MA-32) and DST-SERB grant EMR/2016/004810.
    The third author was supported in part at the Technion by a fellowship of the Israel Council for Higher Education and the Israel Science Foundation (Grant No. 970/15) founded by the Israel Academy of Sciences and Humanities.
  • © Copyright 2018 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 371 (2019), 4377-4409
  • MSC (2010): Primary 35J15; Secondary 35A02, 35B40, 35B60
  • DOI: https://doi.org/10.1090/tran/7694
  • MathSciNet review: 3917226