Quantitative unique continuation for real-valued solutions to second order elliptic equations in the plane

In this article, we study a quantitative form of the Landis conjecture on exponential decay for real-valued solutions to second order elliptic equations with variable coefficients in the plane. In particular, we prove the following qualitative form of Landis conjecture, for $W_1, W_2 \in L^{\infty}(\mathbb R^2;\mathbb R^2)$, $V \in L^{\infty}(\mathbb R^2;\mathbb R)$ and $u \in H_{\mathrm{loc}}^{1}(\mathbb R^2)$ a real-valued weak solution to $-\Delta u - \nabla \cdot ( W_1 u ) +W_2 \cdot \nabla u + V u = 0$ in $\mathbb R^2$, satisfying for $\delta>0$, $|u(x)| \leq \exp(- |x|^{1+\delta})$, $x \in \mathbb R^2$, then $u \equiv 0$. Our methodology of proof is inspired by the one recently developed by Logunov, Malinnikova, Nadirashvili, and Nazarov that have treated the equation $-\Delta u + V u = 0$ in $\mathbb R^2$. Nevertheless, several differences and additional difficulties appear. New weak quantitative maximum principles are established for the construction of a positive multiplier in a suitable perforated domain, depending on the nodal set of $u$. The resulted divergence elliptic equation is then transformed into a non-homogeneous $\partial_{\overline{z}}$ equation thanks to a generalization of Stoilow factorization theorem obtained by the theory of quasiconformal mappings, an approximate type Poincar\'e lemma and the use of the Cauchy transform. Finally, a suitable Carleman estimate applied to the operator $\partial_{\overline{z}}$ is the last ingredient of our proof.