The Landis Conjecture for variable coefficient second-order elliptic PDES

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 $\|V\|_{L^\infty({\mathbb R}^2)} \le 1$. 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.