论文标题
线性化calderón问题：无限扰动的无限维空间的重建和Lipschitz稳定性
Linearised Calderón problem: Reconstruction and Lipschitz stability for infinite-dimensional spaces of unbounded perturbations
论文作者
论文摘要
我们研究了一个线性化的calderón问题，在二维有限的简单连接的$ c^{1，α} $域$ω$。在扩展了$ l^2（ω）$扰动的线性问题之后，我们正交分解$ l^2（ω）= \ oplus_ {k = 0}^\ infty \ infty \ mathcal {h} _k $ and co $ $ \ mathcal} $ {特别是，$ \ Mathcal {H} _0 $是方形可谐波扰动的空间。这似乎是在（线性化的）calderón问题的背景下，无限二维空间的第一个Lipschitz稳定性结果。关于数据图的运算符规范的先前最佳估计已在无限维设置中的对数类型。通过使用Hilbert-Schmidt Narm进行Neumann到Dirichlet边界图及其相对于电导率系数的Fréchet导数，可以实现显着的改进。我们还得出了一种直接的重建方法，该方法可将一般$ l^2（ω）$扰动的正交预测到$ \ Mathcal {H} _K $空间上，因此重建任何$ l^2（ω）$ pertturlation。
We investigate a linearised Calderón problem in a two-dimensional bounded simply connected $C^{1,α}$ domain $Ω$. After extending the linearised problem for $L^2(Ω)$ perturbations, we orthogonally decompose $L^2(Ω) = \oplus_{k=0}^\infty \mathcal{H}_k$ and prove Lipschitz stability on each of the infinite-dimensional $\mathcal{H}_k$ subspaces. In particular, $\mathcal{H}_0$ is the space of square-integrable harmonic perturbations. This appears to be the first Lipschitz stability result for infinite-dimensional spaces of perturbations in the context of the (linearised) Calderón problem. Previous optimal estimates with respect to the operator norm of the data map have been of the logarithmic-type in infinite-dimensional settings. The remarkable improvement is enabled by using the Hilbert-Schmidt norm for the Neumann-to-Dirichlet boundary map and its Fréchet derivative with respect to the conductivity coefficient. We also derive a direct reconstruction method that inductively yields the orthogonal projections of a general $L^2(Ω)$ perturbation onto the $\mathcal{H}_k$ spaces, hence reconstructing any $L^2(Ω)$ perturbation.
