Quantitative unique continuation for Schrödinger operators
Quantitative unique continuation for Schrödinger operators
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薛定谔算子的定量唯一延拓
DOI：
10.1016/j.jfa.2020.108566
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发表时间：
2019
影响因子：
1.7
通讯作者：
Blair Davey
中科院分区：
文献类型：
--
作者：
Blair Davey
研究了具有奇异低阶项的二阶椭圆型方程解的定量唯一连续性质。主要定理给出了Δ+ V的强唯一连续性的一个量化，即对于任意非平凡的u，若在Rn的某个开连通子集中解Δ u+ Vu = 0，我们用V的Lt-范数估计解的消失阶。我们的结果适用于所有t> n2和n≥ 3。有了这些最大阶的消失估计，我们使用一个尺度参数来产生Δ u+ V u= 0的全局解在无穷远处的定量唯一连续估计。对于任意t∈（n2，∞]，为了处理V∈ Lt，我们通过用一个新的端点Carleman估计插值一个已知的Lp-L2估计，证明了一个新的Lp-Lq Carleman估计.这个新的Carleman估计也可以用来建立具有一阶项的方程的消失估计的改进阶，这些方程的形式为Δ u+ W u+ V u= 0。
We investigate the quantitative unique continuation properties of solutions to second order elliptic equations with singular lower order terms. The main theorem presents a quantification of the strong unique continuation property for Δ+ V. That is, for any non-trivial u that solves Δ u+ V u= 0 in some open, connected subset of R n, we estimate the vanishing order of solutions in terms of the L t-norm of V. Our results apply to all t> n 2 and n≥ 3. With these maximal order of vanishing estimates, we employ a scaling argument to produce quantitative unique continuation at infinity estimates for global solutions to Δ u+ V u= 0. To handle V∈ L t for every t∈(n 2,∞], we prove a novel L p− L q Carleman estimate by interpolating a known L p− L 2 estimate with a new endpoint Carleman estimate. This new Carleman estimate may also be used to establish improved order of vanishing estimates for equations with a first order term, those of the form Δ u+ W⋅∇ u+ V u= 0.
DOI：
10.1080/03605302.2019.1629957
发表时间：
2017-02
影响因子：
1.9
作者：
Blair Davey;Jiuyi Zhu
通讯作者：
Blair Davey;Jiuyi Zhu