  In this article, we study the order of vanishing and a quantitative form of
Landis' conjecture in the plane for solutions to second-order elliptic
equations with variable coefficients and singular lower order terms. Precisely,
we let $A$ be real-valued, bounded and elliptic, but not necessary symmetric or
continuous, and we assume that $V$ and $W_i$ are real-valued and belong to
$L^p$ and $L^{q_i}$, respectively. We prove that if $u$ is a real-valued,
bounded and normalized solution to an equation of the form $-\nabla \cdot (A
\nabla u + W_1 u) + W_2 \cdot \nabla u + V u = 0$ in $B_d$, then under suitable
conditions on the lower order terms, for any $r$ sufficiently small, the
following order of vanishing estimate holds $$\|u\|_{L^\infty(B_r)} \ge r^{C
M},$$ where $M$ depends on the Lebesgue norms of the lower order terms. In a
number of settings, a scaling argument gives rise to a quantitative form of
Landis' conjecture, \[ \inf_{|z_0| = R} \|u\|_{L^\infty(B_1(z_0))} \ge \exp(- C
R^\beta \log R), \] where $\beta$ depends on $p$, $q_1$, and $q_2$. The
integrability assumptions that we impose on $V$ and $W_i$ are nearly optimal in
view of a scaling argument. We use the theory of elliptic boundary value
problems to establish the existence of positive multipliers associated to the
elliptic equation. Then the proofs rely on transforming the equations to
Beltrami systems and applying a generalization of Hadamard's three-circle
theorem.
