Title: Boundary determination for the Schrödinger equation with unknown embedded obstacles by local data Show Chinese title

Title: 薛定谔方程未知嵌入障碍物的边界确定通过局部数据

Abstract: In this paper, we consider the inverse boundary value problem of the elliptic operator $\Delta+q$ in a fixed region $\Omega\subset\mathbb{R}^3$ with unknown embedded obstacles $D$. In particular, we give a new and simple proof to uniquely determine $q$ and all of its derivatives at the boundary from the knowledge of the local Dirichlet-to-Neumann map on $\partial\Omega$, disregarding the unknown obstacle, where in fact only the local Cauchy data of the fundamental solution is used. Our proof mainly depends on the rigorous singularity analysis on certain singular solutions and the volume potentials of fundamental solution, which is easy to extend to many other cases. Show Chinese abstract

Abstract: 在本文中，我们考虑固定区域 $\Omega\subset\mathbb{R}^3$ 内椭圆算子 $\Delta+q$ 的逆边界值问题，其中包含未知的嵌入障碍物 $D$。 特别是，我们给出了一种新的且简单的证明，以唯一确定 $q$ 及其在边界上的所有导数，仅根据对区域 $\partial\Omega$ 上局部狄利克雷到诺伊曼映射的了解，而不考虑未知障碍物，实际上仅使用了基本解的局部柯西数据。 我们的证明主要依赖于对某些奇异解和基本解的体积势的严格奇异性分析，这很容易扩展到许多其他情况。