论文标题
Gel'Fand的逆边界问题的定量稳定性
Quantitative stability of Gel'fand's inverse boundary problem
论文作者
论文摘要
在Gel'Fand的反问题中，一个人旨在确定紧凑型歧管$ M $的拓扑，差异结构和riemannian指标，并从界限$ \ partial m，$ neumann eigenvalues $λ_j$和特征函数的边界价值和特征函数的边界值$ _j_j_j__j | _ _} $ _} $} $} $} $} $} $} $} $} $} $。我们表明，此问题具有稳定的解决方案，并在具有有界几何形状的一类歧管中具有定量稳定性估计。更确切地说，我们表明，有限的许多特征值和相应特征函数的边界值（已知为小误差）确定了一个与Gromov-Hausdorff Sense中歧管接近的度量空间。我们提供算法来构建此度量空间。该结果基于对波算子唯一延续的稳定性的明确估计。
In Gel'fand's inverse problem, one aims to determine the topology, differential structure and Riemannian metric of a compact manifold $M$ with boundary from the knowledge of the boundary $\partial M,$ the Neumann eigenvalues $λ_j$ and the boundary values of the eigenfunctions $φ_j|_{\partial M}$. We show that this problem has a stable solution with quantitative stability estimates in a class of manifolds with bounded geometry. More precisely, we show that finitely many eigenvalues and the boundary values of corresponding eigenfunctions, known up to small errors, determine a metric space that is close to the manifold in the Gromov-Hausdorff sense. We provide an algorithm to construct this metric space. This result is based on an explicit estimate on the stability of the unique continuation for the wave operator.
