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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…

Analysis of PDEs · Mathematics 2008-09-19 Oleg Yu. Imanuvilov , Gunther Uhlmann , masahiro Yamamoto

For the two dimensional Schr\"odinger equation in a bounded domain, we prove uniqueness of determination of potentials in $W^1_p(\Omega),\,\, p>2$ in the case where we apply all possible Neumann data supported on an arbitrarily non-empty…

Mathematical Physics · Physics 2012-10-05 O. Imanuvilov , G. Uhlmann , M. Yamamoto

In this paper we show, in dimension n >=3, that knowledge of the Cauchy data for the Schroedinger equation in the presence of a magnetic potential, measured on possibly very small subsets of the boundary, determines uniquely the magnetic…

Analysis of PDEs · Mathematics 2007-05-23 David Dos Santos Ferreira , Carlos Kenig , Johannes Sjoestrand , Gunther Uhlmann

We consider the Dirichlet-to-Neumann map associated to the Schr\"odinger equation with a potential in a bounded Lipschitz domain in three or more dimensions. We show that the integral of the potential over a two-plane is determined by the…

Analysis of PDEs · Mathematics 2007-05-23 Allan Greenleaf , Gunther Uhlmann

We show that the knowledge of the set of the Cauchy data on the boundary of a bounded open set in $\R^n$, $n\ge 3$, for the magnetic Schr\"odinger operator with $L^\infty$ magnetic and electric potentials determines the magnetic field and…

Analysis of PDEs · Mathematics 2012-07-10 Katsiaryna Krupchyk , Gunther Uhlmann

We show global uniqueness in an inverse problem for the fractional Schr\"odinger equation: an unknown potential in a bounded domain is uniquely determined by exterior measurements of solutions. We also show global uniqueness in the partial…

Analysis of PDEs · Mathematics 2020-03-25 Tuhin Ghosh , Mikko Salo , Gunther Uhlmann

We consider uniqueness in an inverse Schr\"odinger problem in a bounded domain in $\mathbb{R}^2$ given the Dirichlet-to-Neumann map on part of the boundary. On the remaining boundary we impose a new type of singular boundary condition with…

Analysis of PDEs · Mathematics 2018-09-19 Freddy J. F. Symons

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,…

Analysis of PDEs · Mathematics 2022-04-12 Giovanni Covi , Jesse Railo , Philipp Zimmermann

We study the uniqueness question for two inverse problems on graphs. Both problems consist in finding (possibly complex) edge or nodal based quantities from boundary measurements of solutions to the Dirichlet problem associated with a…

Combinatorics · Mathematics 2015-10-13 Justin Boyer , Jack J. Garzella , Fernando Guevara Vasquez

We consider the inverse problem of determining the coefficients of a general second-order elliptic operator in two dimensions by measuring the corresponding Cauchy data on an arbitrary open subset of the boundary. We show that one can…

Analysis of PDEs · Mathematics 2010-10-29 O. Imanuvilov , G. Uhlmann , M. Yamamoto

For a semilinear elliptic equation, we prove uniqueness results in determining potentials and semilinear terms from partial Cauchy data on an arbitrary subboundary.

Mathematical Physics · Physics 2012-05-22 Oleg Imanuvilov , Masahiro Yamamoto

In this paper we improve an earlier result by Bukhgeim and Uhlmann, by showing that in dimension larger than or equal to three, the knowledge of the Cauchy data for the Schr\"odinger equation measured on possibly very small subsets of the…

Analysis of PDEs · Mathematics 2007-05-23 C. E. Kenig , J. Sjoestrand , G. Uhlmann

We prove identification of coefficients up to gauge by Cauchy data at the boundary for elliptic systems on oriented compact surfaces with boundary or domains of $\mathbb{C}$. In the geometric setting, we fix a Riemann surface with boundary,…

Analysis of PDEs · Mathematics 2011-05-24 Pierre Albin , Colin Guillarmou , Leo Tzou , Gunther Uhlmann

We demonstrate a quantitative version of the usual properties related to unique continuation from an interior datum for the Schr\"odinger equation with bounded or unbounded potential. The inequalities we establish have constants that…

Analysis of PDEs · Mathematics 2025-04-11 Mourad Choulli , Hiroshi Takase

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…

Analysis of PDEs · Mathematics 2019-01-11 Giovanni Covi

We show that the knowledge of the set of the Cauchy data on the boundary of a $C^1$ bounded open set in $\R^n$, $n\ge 3$, for the Schr\"odinger operator with continuous magnetic and bounded electric potentials determines the magnetic field…

Analysis of PDEs · Mathematics 2012-07-10 Katsiaryna Krupchyk , Gunther Uhlmann

We prove the global uniqueness in determination of the conductivity, the permeability and the permittivity of two dimensional Maxwell's equations by partial Dirichlet-to-Neumann map limited to an arbitrary subboundary.

Mathematical Physics · Physics 2014-04-01 O. Yu. Imanuvilov M. Yamamoto

The purpose of this article is to extend the uniqueness results for the two dimensional Calder\'on problem to unbounded potentials on general geometric settings. We prove that the Cauchy data sets for Schr\"odinger equations uniquely…

Analysis of PDEs · Mathematics 2020-07-14 Yilin Ma

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…

Analysis of PDEs · Mathematics 2019-10-10 Giovanni S. Alberti , Matteo Santacesaria

The Schr\"odinger equation is considered on the half line with a selfadjoint boundary condition when the potential is real valued, integrable, and has a finite first moment. It is proved that the potential and the two boundary conditions…

Mathematical Physics · Physics 2007-05-23 Tuncay Aktosun , Ricardo Weder
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