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

In this article, we study qualitative and quantitative forms of the Landis' conjecture on exponential decay for real-valued planar solutions to second order elliptic equations with bounded first and zero-order terms.We establish that for anycannot decay faster than exponentially.Our strategy of proof is inspired by the methodology of Logunov, Malinnikova, Nadirashvili, and Nazarov, who have treated zero-order perturbations of the Laplacian.Nevertheless, several differences and additional difficulties arise.New weak quantitative maximum principles are proved for the construction of a positive multiplier in a suitable perforated domain, depending on the nodal set of u.The resulting divergence-form elliptic equation is then transformed into a nonhomogeneous ∂ z equation thanks to a generalization of the Stoilow factorization theorem obtained by the theory of quasiconformal mappings, an approximate type Poincaré lemma and the use of the Cauchy transform.Finally, a suitable Carleman estimate applied to the operator ∂ z is the last ingredient of our proof.