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A partial data result for less regular conductivities in admissible geometries  [PDF]
Casey Rodriguez
Mathematics , 2014,
Abstract: We consider the Calder\'on problem with partial data in certain admissible geometries, that is, on compact Riemannian manifolds with boundary which are conformally embedded in a product of the Euclidean line and a simple manifold. We show that measuring the Dirichlet-to-Neumann map on roughly half of the boundary determines a conductivity that has essentially 3/2 derivatives. As a corollary, we strengthen a partial data result due to Kenig, Sj\"ostrand, and Uhlmann.
The Calderón problem with partial data for conductivities with $3/2$ derivatives  [PDF]
Katya Krupchyk,Gunther Uhlmann
Mathematics , 2015,
Abstract: We extend a global uniqueness result for the Calder\'on problem with partial data, due to Kenig-Sj\"ostrand-Uhlmann, to the case of less regular conductivities. Specifically, we show that in dimensions $n\ge 3$, the knowledge of the Diricihlet-to-Neumann map, measured on possibly very small subsets of the boundary, determines uniquely a conductivity having essentially $3/2$ derivatives in an $L^2$ sense.
Uniqueness in Calderon's problem with Lipschitz conductivities  [PDF]
Boaz Haberman,Daniel Tataru
Mathematics , 2011, DOI: 10.1215/00127094-2019591
Abstract: We use X^{s,b}-inspired spaces to prove a uniqueness result for Calderon's problem in a Lipschitz domain under the assumption that the conductivity is Lipschitz. For Lipschitz conductivities, we obtain uniqueness for conductivities close to the identity in a suitable sense. We also prove uniqueness for arbitrary C^1 conductivities.
Gel'fand-Calderón's inverse problem for anisotropic conductivities on bordered surfaces in $\mathbb{R}^3$  [PDF]
Gennadi Henkin,Matteo Santacesaria
Mathematics , 2010, DOI: 10.1093/imrn/rnr046
Abstract: Let $X$ be a smooth bordered surface in $\real^3$ with smooth boundary and $\hat \sigma$ a smooth anisotropic conductivity on $X$. If the genus of $X$ is given, then starting from the Dirichlet-to-Neumann operator $\Lambda_{\hat \sigma}$ on $\partial X$, we give an explicit procedure to find a unique Riemann surface $Y$ (up to a biholomorphism), an isotropic conductivity $\sigma$ on $Y$ and the boundary values of a quasiconformal diffeomorphism $F: X \to Y$ which transforms $\hat \sigma$ into $\sigma$. As a corollary we obtain the following uniqueness result: if $\sigma_1, \sigma_2$ are two smooth anisotropic conductivities on $X$ with $\Lambda_{\sigma_1}= \Lambda_{\sigma_2}$, then there exists a smooth diffeomorphism $\Phi: \bar X \to \bar X$ which transforms $\sigma_1$ into $\sigma_2$.
Global uniqueness for the Calderón problem with Lipschitz conductivities  [PDF]
Pedro Caro,Keith Rogers
Mathematics , 2014,
Abstract: We prove uniqueness for Calder\'on's problem with Lipschitz conductivities in higher dimensions. Combined with the recent work of Haberman, who treated the three and four dimensional cases, this confirms a conjecture of Uhlmann. Our proof builds on the work of Sylvester and Uhlmann, Brown, and Haberman and Tataru who proved uniqueness for $C^1$ conductivities and Lipschitz conductivities sufficiently close to the identity.
Stability of the Calderón problem for less regular conductivities  [PDF]
Pedro Caro,Andoni García,Juan Manuel Reyes
Mathematics , 2012,
Abstract: In these notes we prove log-type stability for the Calder\'on problem with conductivities in $ C^{1,\varepsilon}(\bar{\Omega}) $. We follow the lines of a recent work by Haberman and Tataru in which they prove uniqueness for $ C^1(\bar{\Omega}) $.
Reconstruction of less regular conductivities in the plane  [PDF]
Kim Knudsen,Alexandru Tamasan
Mathematics , 2001,
Abstract: We study the inverse conductivity problem of how to reconstruct an isotropic electrical conductivity distribution $\gamma$ in an object from static electrical measurements on the boundary of the object. We give an exact reconstruction algorithm for the conductivity $\gamma\in C^{1+\epsilon}(\ol \Om)$ in the plane domain $\Omega$ from the associated Dirichlet to Neumann map on $\partial \Om.$ Hence we improve earlier reconstruction results. The method used relies on a well-known reduction to a first order system, for which the $\ol\partial$-method of inverse scattering theory can be applied.
Uniqueness in Calderón's problem for conductivities with unbounded gradient  [PDF]
Boaz Haberman
Mathematics , 2014, DOI: 10.1007/s00220-015-2460-3
Abstract: We prove uniqueness in the inverse conductivity problem for uniformly elliptic conductivities in $W^{s,p}(\Omega)$, where $\Omega \subset \mathbb R^n$ is Lipschitz, $3\leq n \leq 6$, and $s$ and $p$ are such that $ W^{s,p}(\Omega)\not \subset W^{1,\infty}(\Omega)$. In particular, we obtain uniqueness for conductivities in $W^{1,n}(\Omega)$ ($n=3,4$). This improves on the result of the author and Tataru, who assumed that the conductivity is Lipschitz.
Some rigorous relations for partial conductivities in ionic liquids  [cached]
I. Mryglod,V. Kuporov
Condensed Matter Physics , 2010,
Abstract: Starting with the rigorous expressions, derived previously for the generalized transport coefficients of a multi-component fluid, we obtained several exact relations for partial conductivities of ionic charge-asymmetric mixtures. For a simpler case of a charge-symmetric binary mixture such kind of relations was discovered experimentally by Sundheim more than 50 years ago and is known as the "universal golden rule". Some more complicate models, describing in particular the cases of ternary and multi-component mixtures, are considered. The general relation for partial ionic conductivities is derived for a multi-component ionic fluid. It is shown that such relations can be considered in fact as an example of a more general class of rigorous expressions valid for (k,ω)-dependent quantities.
On the Dirichlet Problem for Backward Parabolic Stochastic Partial Differential Equations in General Smooth Domains  [PDF]
Kai Du,Shanjian Tang
Mathematics , 2009,
Abstract: Backward stochastic partial differential equations of parabolic type with variable coefficients are considered in smooth domains. Existence and uniqueness results are given in weighted Sobolev spaces allowing the derivatives of the solutions to blow up near the boundary.
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