局部共形横观各向异性几何中由边界数据确定势的唯一性
Uniqueness of a Potential from Boundary Data in Locally Conformally Transversally Anisotropic Geometries
摘要：
设$(Ω^3,g)$是具有光滑边界的紧致光滑黎曼流形，且$U$是$Ω$中的开集，使得$g|_U$是欧氏度量.设$Γ= \overline{U} \cap \partial Ω$是连通的，并假设$U$是$Γ$的凸包。我们将研究薛定谔算子$ -\triangle_g + q $从相应的Dirichlet到Neumann映射$Λ_q$的未知势的唯一性。我们将证明，如果潜在的$q$是一个先验明确已知的$U^c$，然后可以唯一地重建$q$的凸包$Γ$从$Λ_q$。我们还将概述一个重建算法。更一般地，我们将讨论的情况下，$Γ$是不连通的或$g|_{U}$是共形横向各向异性，并得出类似的结果。
英文摘要：
Let $(Ω^3,g)$ be a compact smooth Riemannian manifold with smooth boundary and suppose that $U$ is a an open set in $Ω$ such that $g|_U$ is the Euclidean metric. Let $Γ= \overline{U} \cap \partial Ω$ be connected and suppose that $U$ is the convex hull of $Γ$. We will study the uniqueness of an unknown potential for the Schrödinger operator $ -\triangle_g + q $ from the associated Dirichlet to Neumann map, $Λ_q$. We will prove that if the potential $q$ is a priori explicitly known in $U^c$, then one can uniquely reconstruct $q$ over the convex hull of $Γ$ from $Λ_q$. We will also outline a reconstruction algorithm. More generally we will discuss the cases where $Γ$ is not connected or $g|_{U}$ is conformally transversally anisotropic and derive the analogous result.
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