The Landis Conjecture for variable coefficient second-order elliptic PDES

Abstract

In this work, we study the Landis conjecture for second-order elliptic equations in the plane. Precisely, assume that $V\ge 0$ is a measurable real-valued function satisfying $\Vert V{\Vert }_{L^{\infty }({\mathbb{R}}^2)}\le 1$. Let $u$ be a real solution to $\mbox{div}(A \nabla u) - V u = 0$ in ${\mathbb{R}}^2$. Assume that $|u(z)|\le \exp (c_0|z|)$ and $u(0)=1$. Then, for any $R$ sufficiently large, \[ \inf_{|z_0| = R} \|u\|_{L^\infty(B_1(z_0))} \ge \exp(- C R \log R). \] 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-quasi-circle theorem.

References in corpus (1)

• Some quantitative unique continuation results for eigenfunctions of the magnetic Schrödinger operator

Cited by in corpus (2)

• Landis-type conjecture for the half-Laplacian
• Gaussian decay of Harmonic Oscillators and related models