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Transactions of the American Mathematical Society

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Certain Liouville properties of eigenfunctions of elliptic operators


Authors: Ari Arapostathis, Anup Biswas and Debdip Ganguly
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
Published electronically: November 26, 2018
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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
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
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
Email: debdip@iiserpune.ac.in

DOI: https://doi.org/10.1090/tran/7694
Keywords: Principal eigenvalue, the Landis conjecture, decay of eigenfunctions, Liouville property
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.
Article copyright: © Copyright 2018 American Mathematical Society