The Landis conjecture for variable coefficient second-order elliptic PDEs
Journal
Transactions of the American Mathematical Society
Journal Volume
369
Journal Issue
11
Pages
8209-8237
Date Issued
2017
Author(s)
Abstract
In this article, we study the quantitative form of Landis’ conjecture in the plane for second-order elliptic equations with variable coefficients. Precisely, let A be a symmetric, positive-definite matrix with Lipschitz coefficients. Assume that V ≥ 0 is a measurable, real-valued function satisfying
V
L∞(ℝ2 ) ≤ 1. Let u be a real-valued solution to div(A∇u) − Vu =0 in ℝ2. If u is bounded and normalized in the sense that |u(z)| ≤exp(c0 |z|) and u(0) = 1, then for any R sufficiently large, (Formula presented) In addition to equations with electric potentials, we also derive similar estimates for equations with first-order terms, or magnetic potentials. The proofs rely on transforming the equations to Beltrami systems and applying a generalization of Hadamard’s three-circle theorem.
Type
journal article