Series reversion in Calderón's problem

Uniqueness in the Calderón problem via infinitesimally bounded potentials

The Calderón problem is an inverse problem with applications to electrical impedance tomography and geophysical prospection. We prove uniqueness in the Calderón problem in spatial dimension $n \geq 3$ for scalar conductivities in the Sobolev space $W^{1,p}$ with $p \geq n$. This generalizes a result of Haberman who considered the case $p \geq n$ and $n=3$ or $4$. Our method of proof combines a Fourier series approach with an analytic criterion for infinitesimal boundedness of potentials appearing in a Schrödinger equation with respect to the Laplacian.

分数阶Calderón问题中局部边界数据的边界确定

Boundary determination from local boundary data for a fractional Calderón problem

针对分数阶Schrödinger算子的局部边界条件，引入新Calderón型问题，证明Dirichlet-to-Neumann算子可确定边界处势的泰勒级数，从而唯一确定解析势。

A Uniqueness Result for the Calderon Problem for $U(N)$-connections coupled to spinors

In this paper we define a Dirichlet-to-Neumann map for a twisted Dirac Laplacian acting on bundle-valued spinors over a spin manifold. We show that this map is a pseudodifferential operator of order 1 whose symbol determines the Taylor series of the metric and connection at the boundary. We go on to show that if two real-analytic connections couple to a spinor via the Yang--Mills--Dirac equations with appropriate boundary conditions, and have equal Dirichlet-to-Neumann maps, then the two connections are locally gauge equivalent. In the abelian case, the connections are globally gauge equivalen…

Calderón problem for Yang-Mills connections

We consider the problem of identifying a unitary Yang-Mills connection $\nabla$ on a Hermitian vector bundle from the Dirichlet-to-Neumann (DN) map of the connection Laplacian $\nabla^*\nabla$ over compact Riemannian manifolds with boundary. We establish uniqueness of the connection up to a gauge equivalence in the case of trivial line bundles in the smooth category and for the higher rank case in the analytic category, by using geometric analysis methods and essentially only one measurement. Moreover, by using a Runge-type approximation argument along curves to recover holonomy, we are able t…

Beltrami场的Calderón问题

A Calderón Problem for Beltrami Fields

本文研究3维黎曼流形上Beltrami场的法向到切向映射，证明该零阶伪微分算子的全符号决定边界度量Taylor级数，并实现实解析单连通流形的重建。

带有手性边界条件的Dirac算子的Calderón问题

A Calderón Problem for the Dirac operator with chiral boundary conditions

本文研究了带有手性边界条件的Dirac算子的Calderón问题，证明了边界共轭映射能唯一确定流形和联络的结构。