外区域中的非线性薛定谔方程
Nonlinear Schrödinger equation in an exterior domain
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中文总结 AI 辅助
该研究针对非捕获障碍物外区域的三次、五次非线性薛定谔方程，利用局部光滑效应与Strichartz型估计，证明了其在二维、三维中围绕零解及非平凡轨迹的局部精确可控性，并讨论了相关开放问题。
中文摘要 AI 辅助
我们研究非捕获障碍物Θ的外区域Ω=ℝⁿ\backslashΘ中三次非线性薛定谔方程的精确可控性，该方程具有支撑在大球外部的内部控制，以及在Ω₀=B_{R₀}\backslashΘ上仅作用于外球面∂B_{R₀}的狄利克雷边界控制问题。利用Burq、Gérard和Tzvetkov[9]的局部光滑效应，我们证明了与狄利克雷拉普拉斯算子相关的整个索伯列夫空间尺度H^σ_D(Ω)（σ∈[-2,2]）上线性薛定谔方程的可观性不等式；这些不等式仅要求非捕获假设（无需Θ为星形区域），且其证明不使用法向边界迹。结合Strichartz型估计和扰动论证，这在二维和三维中得到了围绕零解的H¹₀(Ω)和H²(Ω)∩H¹₀(Ω)的局部精确可控性，对于内部控制且在唯一延拓假设下，还可得到围绕非平凡轨迹的局部精确可控性。相同方法适用于五次方程，该方程在三维中是能量临界的。文中还讨论了若干开放问题。
英文摘要
We study the exact controllability of the cubic nonlinear Schrödinger equation in the exterior $Ω=\mathbb{R}^n\setminusΘ$ of a non-trapping obstacle $Θ$, with internal controls supported outside a large ball, and the corresponding boundary control problem on $Ω_0=B_{R_0}\setminusΘ$ with Dirichlet controls acting only on the outer sphere $\partial B_{R_0}$. Using the local smoothing effect of Burq, Gérard and Tzvetkov [9], we prove observability inequalities for the linear Schrödinger equation in the whole scale of Sobolev spaces $H^σ_D(Ω)$, $σ\in[-2,2]$, associated with the Dirichlet Laplacian; they require only the non-trapping assumption (no star-shapedness of $Θ$), and their proofs use no normal boundary traces. Combined with Strichartz-type estimates and a perturbation argument, this yields local exact controllability in $H^1_0(Ω)$ and in $H^2(Ω)\cap H^1_0(Ω)$, in dimensions two and three, around the zero solution and, for internal controls and under a unique continuation assumption, around nontrivial trajectories. The same method applies to the quintic equation, which is energy-critical in dimension three. Several open problems are discussed.
发表机构
- Universidade Federal de Pernambuco（巴西联邦伯南布哥大学）
- University of Cincinnati（辛辛那提大学）
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