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In this paper we study the inverse conductivity problem with partial data. Moreover, we show that, in dimension $n\geq 3$ the uniqueness of the Calder\'{o}n problem holds for the $C^{1}\bigcap H^{3/2, 2}$ conductivities.

偏微分方程分析 · 数学 2015-06-05 Guo Zhang

We consider the inverse conductivity problem in a strictly convex domain whose boundary is not known. Usually the numerical reconstruction from the measured current and voltage data is done assuming the domain has a known fixed geometry.…

偏微分方程分析 · 数学 2016-09-07 Ville Kolehmainen , Matti Lassas , Petri Ola

We consider the inverse conductivity problem of identifying embedded objects in unbounded domains. The main tool is a set of special solutions to the Schroedinger equation, the complex spherical waves, which are constructed by a Carleman…

偏微分方程分析 · 数学 2009-11-11 Mikko Salo , Jenn-Nan Wang

We prove uniqueness of the inverse conductivity problem in three dimensions for complex conductivities in $W^{1,\infty}$. We apply quaternionic analysis to transform the inverse problem into an inverse Dirac scattering problem, as…

偏微分方程分析 · 数学 2023-01-23 Ivan Pombo

We construct anisotropic conductivities with the same Dirichlet-to-Neumann map as a homogeneous isotropic conductivity. These conductivities are singular close to a surface inside the body.

偏微分方程分析 · 数学 2007-05-23 Allan Greenleaf , Matti Lassas , Gunther Uhlmann

We study inverse conductivity problem for an anisotropic conductivity in $L^\infty$ in bounded and unbounded domains. Also, we give applications of the results in the case when Dirichlet-to-Neumann and Neumann-to-Dirichlet maps are given…

偏微分方程分析 · 数学 2007-05-23 Kari Astala , Matti Lassas , Lassi Paivarinta

Electrical Impedance Imaging would suffer a serious obstruction if for two different conductivities the potential and current measured at the boundary were the same. The Calder\'on's problem is to decide whether the conductivity is indeed…

偏微分方程分析 · 数学 2021-09-21 Felipe Ponce-Vanegas

We consider the inverse Calder\'on problem consisting of determining the conductivity inside a medium by electrical measurements on its surface. Ideally, these measurements determine the Dirichlet-to-Neumann map and, therefore, one usually…

偏微分方程分析 · 数学 2017-06-28 Pedro Caro , Andoni Garcia

This paper shows global uniqueness in two inverse problems for a fractional conductivity equation: an unknown conductivity in a bounded domain is uniquely determined by measurements of solutions taken in arbitrary open, possibly disjoint…

偏微分方程分析 · 数学 2019-01-11 Giovanni Covi

We generalize many recent uniqueness results on the fractional Calder\'on problem to cover the cases of all domains with nonempty exterior. The highlight of our work is the characterization of uniqueness and nonuniqueness of partial data…

偏微分方程分析 · 数学 2024-09-10 Jesse Railo , Philipp Zimmermann

We prove \emph{global} uniqueness for an inverse problem for the fractional conductivity equation on domains that are bounded in one direction. The conductivities are assumed to be isotropic and nontrivial in the exterior of the domain,…

偏微分方程分析 · 数学 2022-04-12 Giovanni Covi , Jesse Railo , Philipp Zimmermann

We characterize partial data uniqueness for the inverse fractional conductivity problem with $H^{s,n/s}$ regularity assumptions in all dimensions. This extends the earlier results for $H^{2s,\frac{n}{2s}}\cap H^s$ conductivities by Covi and…

偏微分方程分析 · 数学 2024-09-10 Jesse Railo , Philipp Zimmermann

The unique determination of electrical conductivity is extensively studied for isotropic conductivity ever since Calderon's suggestion of the EIT (Electrical Impedance Tomography) problem. However, it is known that there are many…

偏微分方程分析 · 数学 2013-04-25 Kiwoon Kwon

We prove a global uniqueness result for the Calder\'{o}n inverse problem for a general quasilinear isotropic conductivity equation on a bounded open set with smooth boundary in dimension $n\ge 3$. Performing higher order linearizations of…

偏微分方程分析 · 数学 2023-05-10 Cătălin I. Cârstea , Ali Feizmohammadi , Yavar Kian , Katya Krupchyk , Gunther Uhlmann

We show in two dimensions that measuring Dirichlet data for the conductivity equation on an open subset of the boundary and, roughly speaking, Neumann data in slightly larger set than the complement uniquely determines the conductivity on a…

偏微分方程分析 · 数学 2008-09-19 Oleg Yu. Imanuvilov , Gunther Uhlmann , masahiro Yamamoto

We prove that an $L^\infty$ potential in the Schr\"odinger equation in three and higher dimensions can be uniquely determined from a finite number of boundary measurements, provided it belongs to a known finite dimensional subspace…

偏微分方程分析 · 数学 2019-10-10 Giovanni S. Alberti , Matteo Santacesaria

We are concerned with the Calder\'on inverse inclusion problem, where one intends to recover the shape of an inhomogeneous conductive inclusion embedded in a homogeneous conductivity by the associated boundary measurements. We consider the…

偏微分方程分析 · 数学 2021-05-26 Hongyu Liu , Chun-Hsiang Tsou , Wei Yang

In this paper, we consider the inverse scattering problem associated with an inhomogeneous media with a conductive boundary. First, we discuss the inverse conductivity problem of reconstructing the conductivity parameter from scattering…

偏微分方程分析 · 数学 2017-12-12 Isaac Harris , Andreas Kleefeld

We investigate two-dimensional invisibility cloaking via transformation optics approach. The cloaking media possess much more singular parameters than those having been considered for three-dimensional cloaking in literature. Finite energy…

偏微分方程分析 · 数学 2010-06-08 Hongyu Liu , Ting Zhou

We prove for a two dimensional bounded domain that the Cauchy data for the Schroedinger equation measured on an arbitrary open subset of the boundary determines uniquely the potential. This implies, for the conductivity equation, that if we…

偏微分方程分析 · 数学 2008-10-14 Oleg Y. Imanuvilov , Gunther Uhlmann , Masahiro Yamamoto
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