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arXiv 2401.00441math.APmath.CAmath.CV
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平面上二阶椭圆方程实值解的定量唯一延拓性

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

Kévin Le Balc'h, Diego A. Souza

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AI总结：

本文研究了平面上变系数二阶椭圆方程实值解指数衰减的 Landis 猜想的定量形式，通过建立弱定量最大值原理、利用拟共形映射和 Cauchy 变换转化方程，并结合 Carleman 估计，证明了满足特定指数衰减条件的解必为零。

摘要：

在本文中，我们研究了平面上具有变系数的二阶椭圆方程实值解指数衰减的 Landis 猜想的定量形式。特别地，我们证明了以下 Landis 猜想的定性形式：对于 $W_1, W_2 \in L^{\infty}(\mathbb R^2;\mathbb R^2)$，$V \in L^{\infty}(\mathbb R^2;\mathbb R)$ 以及 $u \in H_{\mathrm{loc}}^{1}(\mathbb R^2)$ 为 $\mathbb R^2$ 中 $-Δu - \nabla \cdot ( W_1 u ) +W_2 \cdot \nabla u + V u = 0$ 的实值弱解，若满足对于 $δ>0$，$|u(x)| \leq \exp(- |x|^{1+δ})$，$x \in \mathbb R^2$，则 $u \equiv 0$。我们的证明方法受启发于 Logunov, Malinnikova, Nadirashvili 和 Nazarov 近期发展的方法，他们处理了 $\mathbb R^2$ 中的方程 $-Δu + V u = 0$。然而，出现了若干差异和额外的困难。我们建立了新的弱定量最大值原理，用于在依赖于 $u$ 的节点集的合适穿孔区域中构造正乘子。随后，借助拟共形映射理论得到的 Stoilow 因子分解定理的推广、近似型 Poincaré 引理以及 Cauchy 变换，将所得的散度型椭圆方程转化为非齐次 $\partial_{\overline{z}}$ 方程。最后，对算子 $\partial_{\overline{z}}$ 应用合适的 Carleman 估计是我们证明的最后要素。

英文摘要：

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 $-Δu - \nabla \cdot ( W_1 u ) +W_2 \cdot \nabla u + V u = 0$ in $\mathbb R^2$, satisfying for $δ>0$, $|u(x)| \leq \exp(- |x|^{1+δ})$, $x \in \mathbb R^2$, 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 $-Δu + V u = 0$ 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 Poincaré 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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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