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

Le Balc’h, Kévin; Souza, Diego A.

doi:10.2140/apde.2026.19.1061

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

Kévin Le Balc’h and Diego A. Souza

Vol. 19 (2026), No. 6, 1061–1106

DOI: 10.2140/apde.2026.19.1061

Abstract

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 any $W_{1}, W_{2} \in L^{\infty} (ℝ^{2}; ℝ^{2})$ , $V \in L^{\infty} (ℝ^{2}; ℝ)$ with $∥ W_{1} ∥_{\infty} \leq 1$ , $∥ W_{2} ∥_{\infty} \leq 1$ , $∥ V ∥_{\infty} \leq 1$ and a real-valued solution $u$ to $- Δ u - \nabla \cdot (W_{1} u) + W_{2} \cdot \nabla u + V u = 0$ in $ℝ^{2}$ satisfying $∥ u ∥_{\infty} = | u (0) | = 1$ then for all $δ > 0$ , there exists $C > 0$ such that

\inf_{| x_{0} | = R} \sup_{B (x_{0}, 1)} | u (x) | \geq \exp (- C R^{1 + δ}) for all R \geq 2 .

In particular, real-valued nontrivial planar solutions to $- Δ u - \nabla \cdot (W_{1} u) + W_{2} \cdot \nabla u + V u = 0$ in $ℝ^{2}$ cannot 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 $\partial_{\bar{z}}$ 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 $\partial_{\bar{z}}$ is the last ingredient of our proof.

Keywords

elliptic PDE, unique continuation, Landis conjecture, complex analysis

Mathematical Subject Classification

Primary: 35B60

Secondary: 30C62, 35J15

Milestones

Received: 28 January 2024

Revised: 21 March 2025

Accepted: 27 August 2025

Published: 13 July 2026

Authors

Kévin Le Balc’h
Laboratoire Jacques-Louis Lions Inria Paris, Sorbonne Université Paris France

Diego A. Souza
IMUS and Department of Differential Equations and Numerical Analysis Faculty of Mathematics Universidad de Sevilla Spain

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Vol. 19, No. 6, 2026