The Landis conjecture for variable coefficient second-order elliptic PDEs

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 $V\ge 0$ is a measurable, real-valued function satisfying $\left \vert \left \vert V\right \vert \right \vert _{L^\infty (\mathbb {R}^2)} \le 1$. Let $u$ be a real-valued solution to $\operatorname {div} (A \nabla u) - V u = 0$ in $\mathbb {R}^2$. If $u$ is bounded and normalized in the sense that $\left \vert u(z)\right \vert \le \exp (c_0 |z|)$ 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 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.