On Landis’ conjecture in the plane when the potential has an exponentially decaying negative part
On Landis’ conjecture in the plane when the potential has an exponentially decaying negative part
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DOI：
10.1090/spmj/1600
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发表时间：
2018-08
期刊：
影响因子：
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通讯作者：
Blair Davey;C. Kenig;Jenn-Nan Wang
中科院分区：
文献类型：
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作者：
Blair Davey;C. Kenig;Jenn-Nan Wang
在这篇文章中，我们继续我们的调查在平面上的椭圆方程的实值解的唯一连续性。更准确地说，我们通过建立形式为$- \Delta u + V u = 0$ in $\mathbb{R}^2$的方程的解在无穷远处的唯一延拓估计，进一步证明了兰迪斯猜想的定量版本，其中$V = V_+ - V_-$，$V_+ \in L^\infty$，$V_-$是一个在无穷远处呈现指数衰减的非平凡函数。证明该定理的主要工具是结合迭代方案的消失估计的阶。为了证明消失估计的阶，我们建立了向量值Beltrami系统的相似原理。
In this article, we continue our investigation into the unique continuation properties of real-valued solutions to elliptic equations in the plane. More precisely, we make another step towards proving a quantitative version of Landis' conjecture by establishing unique continuation at infinity estimates for solutions to equations of the form $- \Delta u + V u = 0$ in $\mathbb{R}^2$, where $V = V_+ - V_-$, $V_+ \in L^\infty$, and $V_-$ is a non-trivial function that exhibits exponential decay at infinity. The main tool in the proof of this theorem is an order of vanishing estimate in combination with an iteration scheme. To prove the order of vanishing estimate, we establish a similarity principle for vector-valued Beltrami systems.