势函数反问题中低傅里叶模的改进稳定性
Improved stability of low Fourier modes in inverse problems for potentials
浏览论文内容
中文总结 AI 辅助
该研究针对势函数反问题，基于Dirichlet-to-Neumann映射，分三种正则性情形证明了低傅里叶模恢复的Lipschitz、次Hölder与Hölder稳定性，且势函数无需额外假设，估计常数关于恢复模数量一致。
中文摘要 AI 辅助
我们证明了从Dirichlet-to-Neumann（DN）映射恢复未知势函数低傅里叶模的Lipschitz、次Hölder与Hölder稳定性估计。根据差值$q_1-q_2$的正则性，我们研究了三种不同情形。首先，我们考虑方程 \n\\[ (-Δ- λ^2 + q)u=0 \quad \text{在} \quad Ω\subset \mathbb{R}^n \text{内}. \\]\n我们证明，当差值$q_1-q_2$为$M$带限时，可通过对应DN映射的差值以Lipschitz稳定的方式恢复该差值。只要$λ$相对于$M^{n/2}$足够大，该结论就成立。证明过程用到了实几何光学解。其次，我们考虑方程\n\\[ (-Δ+ q)u=0 \quad \text{在} \quad (-π,π)^n \text{内}. \\]\n我们证明，当差值$q_1-q_2$为实解析且周期函数时，其低傅里叶系数可通过对应DN映射的差值以次Hölder稳定的方式恢复。可恢复的傅里叶模数量随DN映射的接近程度增加而增长。最后，我们考虑差值$q_1-q_2$的傅里叶系数以超指数速率$e^{-c|k|^{n/2}}$衰减的情形。我们证明低傅里叶模可通过Hölder稳定性恢复，且可恢复模的数量随DN映射的接近程度增加而趋于无穷。在所有情形中，$L^\infty$势函数本身无需满足额外假设，也不必属于有限维空间。稳定性估计中的常数关于恢复的傅里叶模数量是一致的。
英文摘要
We prove Lipschitz, sub-Hölder and Hölder stability estimates for recovering the low Fourier modes of an unknown potential from the Dirichlet-to-Neumann (DN) map. We study three different cases, depending on the regularity of the difference $q_1-q_2$. First, we consider \[ (-Δ- λ^2 + q)u=0 \quad \text{in} \quad Ω\subset \mathbb{R}^n. \] We show that the difference $q_1-q_2$, assumed to be $M$-bandlimited, can be recovered in a Lipschitz stable way from the difference of the corresponding DN maps. This holds whenever $λ$ is sufficiently large relative to $M^{n/2}$. The proof involves real geometrical optics solutions. Secondly, we consider \[ (-Δ+ q)u=0 \quad \text{in} \quad (-π,π)^n. \] We show that the low Fourier coefficients of the difference $q_1-q_2$, assumed to be real-analytic and periodic, can be recovered in a sub-Hölder stable way from the difference of the corresponding DN maps. The number of recoverable Fourier modes grows as the DN maps become closer. Finally, we consider the case where the Fourier coefficients of the difference $q_1-q_2$ decay at a super-exponential rate $e^{-c|k|^{n/2}}$. We prove that the low Fourier modes can be recovered with Hölder stability, with the number of recoverable modes tending to infinity as the DN maps become closer. In all cases the $L^\infty$ potentials themselves do not need to satisfy additional assumptions or belong to a finite dimensional space. The constants in the stability estimates are uniform in the number of recovered Fourier modes.
