2012/04/02 by Nuutti Hyvönen, Hyvönen, Nuutti, Petteri Piiroinen +3
Computer Science · Decision Sciences · Mathematics · #35Q60 #35R30 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Probabilistic and Robust Engineering Design #math.AP #msc:35Q60 #msc:35R30
paper · pdf · doi:10.48550/arxiv.1204.0346
arxiv created 2012/04/02 · openalex publication_date 2012/04/02 · arxiv updated 2012/04/03 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28
This work considers properties of the Neumann-to-Dirichlet map for the conductivity equation under the assumption that the conductivity is identically one close to the boundary of the examined smooth, bounded and simply connected domain. It is demonstrated that the so-called bisweep data, i.e., the (relative) potential differences between two boundary points when delta currents of opposite signs are applied at the very same points, uniquely determine the whole Neumann-to-Dirichlet map. In two dimensions, the bisweep data extend as a holomorphic function of two variables to some (interior) neighborhood of the product boundary. It follows that the whole Neumann-to-Dirichlet map is characterized by the derivatives of the bisweep data at an arbitrary point. On the diagonal of the product boundary, these derivatives can be given with the help of the derivatives of the (relative) boundary potentials at some fixed point caused by the distributional current densities supported at the same point, and thus such point measurements uniquely define the Neumann-to-Dirichlet map. This observation also leads to a new, truly local uniqueness result for the so-called Calderón inverse conductivity problem.