Certain Liouville properties of eigenfunctions of elliptic operators

Abstract

We present certain Liouville properties of eigenfunctions of secondorder elliptic operators with real coefficients, via an approach that is based on stochastic representations of positive solutions, and criticality theory of secondorder 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), no. 1, 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.

On Landis’ conjecture in the plane for some equations with sign-changing potentials

“…In [ABG19] and [Ros18], the authors proved qualitative unique continuation at infinity estimates for solutions to elliptic equations under the assumption that the principle eigenvalue is nonnegative. Theorem 1 was motivated by these works and may be interpreted as a quantitative version of their estimates.…”

The Vázquez maximum principle and the Landis conjecture for elliptic PDE with unbounded coefficients

“…The proof of the non-radial case in [44] relies heavily on the fact that e −M |x| is a subsolution of the operator for sufficiently large M > 0, a property which holds only if the coefficients of the operator are bounded. Variants of some of the results in [44] are obtained among other things in the earlier paper [1] via probability techniques, and in the recent work [3] via a duality argument (due to M. Pierre).…”

The Landis conjecture with sharp rate of decay

“…In the case of equations set in the whole space R 2 , the authors are able to handle more general uniformly elliptic operators, still assuming V ≤ 0, see also [7]. As observed in [2], for equations in the whole space, this hypothesis implies that u ≡ 0 just assuming that u ≺ 1, as an immediate consequence of the maximum principle. We point out that the results of [13,7] are deduced from a quantitative estimate which implies that the set where a nontrivial solution is bounded from below by e −h|x|(log |x|) 2 is relatively dense in R 2 .…”

“…The question is motivated by the trivial observation that in dimension N = 1 with V constant, decaying solutions can only exist if V < 0, and they decay as exp(− |V ||x|). Hence, in such case, one can even take κ = sup |V | in condition (2). This is no longer true in higher dimension: the bounded, radial solution of ∆u−u = 0 outside a ball, which can be expressed in terms of the modified Bessel function of second kind, decays like |x| − N −1 2 e −|x| .…”