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

• Main Authors: Balc'h, Kévin Le, Souza, Diego A
• Format: Journal Article
• Language: English
• Published: 31-12-2023
• Subjects: Mathematics - Analysis of PDEs; Mathematics - Classical Analysis and ODEs; Mathematics - Complex Variables
• DOI: 10.48550/arxiv.2401.00441

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.