The Landis conjecture for variable coefficient second-order elliptic PDEs

Abstract

ABSTRACT. In this work, we study the Landis conjecture for second-order elliptic equations in the plane. Precisely, assume that V ≥ 0 is a measurable real-valued function satisfying ||V || L ∞ (R 2 ) ≤ 1. Let u be a real solution to div (A∇u) −Vu = 0 in R 2 . Assume that |u (z)| ≤ exp (c 0 |z|) and u (0) = 1. Then, for any R sufficiently large,In addition to equations with electric potentials, we also derive similar estimates for equations with magnetic potentials. The proofs rely on transforming the equations to Beltrami systems and Hadamard's three-quasicircle theorem.

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

“…Remark 4. Note that these definitions of σ and ρ differ from those that we introduced in [DKW17] and [DW17]. In those papers, the bounds were established over a collection of operators with common ellipticity and boundedness conditions, whereas we are now defining them for a single operator.…”

“…The Beltrami operators are reviewed in Section 5. Since we are able to make some simplifying assumptions for our setting, the results presented here are much simpler than those that previously appeared in [DKW17] and [DW17]. The proof of Theorem 3 is contained in Section 6.…”

“…As is standard, we use the notation B r (z) to denote a ball of radius r centered at z, and occasionally write B r when the center of the ball is understood. Recall from [DKW17] and [DW17] that Q s denotes a quasi-ball associated to a second-order elliptic operator in divergence form, L. More precisely, Q s is a set in R 2 whose boundary curve is an s-level set of the fundamental solution of L. To accommodate for scaling considerations, Q s is defined so that if L = ∆, then Q s = B s . Further details can be found in Section 4.…”

“…The methods from [KSW15] have been used and generalized in subsequent years to prove many Landis-type theorems. In [DKW17], we proved variable-coefficient versions of the theorems in [KSW15] through the use of quasi-conformal transformations. The results in [DW17] apply to very general elliptic equations and rely on the theory of boundary value problems to produce positive multipliers.…”

“…Fundamental solutions and quasi-balls are introduced in Section 4. While the first part of this section is reproduced from [DKW17] and [DW17], the second part provides more refined bounds on the inner and outer radii of our quasi-balls. The Beltrami operators are reviewed in Section 5.…”