Certain Liouville properties of eigenfunctions of elliptic operators

Arapostathis, Ari; Biswas, Anup; Ganguly, Debdip

doi:10.1090/tran/7694

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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