# Reconstruction in the Calder\'on problem on conformally transversally anisotropic manifolds

@article{Feizmohammadi2020ReconstructionIT, title={Reconstruction in the Calder\'on problem on conformally transversally anisotropic manifolds}, author={Ali Feizmohammadi and Katya Krupchyk and Lauri Oksanen and Gunther Uhlmann}, journal={arXiv: Analysis of PDEs}, year={2020} }

We show that a continuous potential $q$ can be constructively determined from the knowledge of the Dirichlet-to-Neumann map for the Schrodinger operator $-\Delta_g+q$ on a conformally transversally anisotropic manifold of dimension $\geq 3$, provided that the geodesic ray transform on the transversal manifold is constructively invertible. This is a constructive counterpart of the uniqueness result of Dos Santos Ferreira-Kurylev-Lassas-Salo. A crucial role in our reconstruction procedure is… Expand

#### 3 Citations

Reconstructing a potential perturbation of the biharmonic operator on transversally anisotropic manifolds

- Mathematics, Physics
- 2021

We prove that a continuous potential q can be constructively determined from the knowledge of the Dirichlet–to–Neumann map for the perturbed biharmonic operator ∆ g + q on a conformally transversally… Expand

Inverse Boundary Problems for Biharmonic Operators in Transversally Anisotropic Geometries

- Mathematics
- SIAM Journal on Mathematical Analysis
- 2021

We study inverse boundary problems for first order perturbations of the biharmonic operator on a conformally transversally anisotropic Riemannian manifold of dimension n ≥ 3. We show that a… Expand

A remark on inverse problems for nonlinear magnetic Schr\"odinger equations on complex manifolds

- Mathematics
- 2021

We show that the knowledge of the Dirichlet–to–Neumann map for a nonlinear magnetic Schrödinger operator on the boundary of a compact complex manifold, equipped with a Kähler metric and admitting… Expand

#### References

SHOWING 1-10 OF 56 REFERENCES

Reconstruction in the partial data Calder\'on problem on admissible manifolds

- Mathematics
- 2015

We consider the problem of developing a method to reconstruct a potential $q$ from the partial data Dirichlet-to-Neumann map for the Schr\"odinger equation $(-\Delta_g+q)u=0$ on a fixed admissible… Expand

Reconstructions from boundary measurements on admissible manifolds

- Mathematics
- 2010

We prove that a potential $q$ can be reconstructed from the Dirichlet-to-Neumann map for the Schrodinger operator $-\Delta_g + q$ in a fixed admissible 3-dimensional Riemannian manifold $(M,g)$. We… Expand

The Calderón problem in transversally anisotropic geometries

- Mathematics
- 2013

We consider the anisotropic Calderon problem of recovering a conductivity matrix or a Riemannian metric from electrical boundary measurements in three and higher dimensions. In the earlier work… Expand

Linearized Calder\'on problem and exponentially accurate quasimodes for analytic manifolds

- Mathematics, Physics
- 2020

In this article we study the linearized anisotropic Calderon problem on a compact Riemannian manifold with boundary. This problem amounts to showing that products of pairs of harmonic functions of… Expand

Inverse Problems for Magnetic Schrödinger Operators in Transversally Anisotropic Geometries

- Physics, Mathematics
- Communications in Mathematical Physics
- 2018

We study inverse boundary problems for magnetic Schrödinger operators on a compact Riemannian manifold with boundary of dimension ≥ 3. In the first part of the paper, we are concerned with the case… Expand

The Inverse Problem for the Dirichlet-to-Neumann map on Lorentzian manifolds

- Mathematics
- 2016

We consider the Dirichlet-to-Neumann map $\Lambda$ on a cylinder-like Lorentzian manifold related to the wave equation related to the metric $g$, a magnetic field $A$ and a potential $q$. We show… Expand

Reconstruction of the magnetic field for a Schr\"odinger operator in a cylindrical setting.

- Mathematics
- 2019

In this thesis we consider a magnetic Schrodinger inverse problem over a compact domain contained in an infinite cylindrical manifold. We show that, under certain conditions on the electromagnetic… Expand

The Calderón problem for the conformal Laplacian

- Mathematics
- 2016

We consider a conformally invariant version of the Calder\'on problem, where the objective is to determine the conformal class of a Riemannian manifold with boundary from the Dirichlet-to-Neumann map… Expand

On determining a Riemannian manifold from the Dirichlet-to-Neumann map

- Mathematics
- 2001

Abstract We study the inverse problem of determining a Riemannian manifold from the boundary data of harmonic functions. This problem arises in electrical impedance tomography, where one tries to… Expand

On reconstruction formulas for the ray transform acting on symmetric differentials on surfaces

- Mathematics
- 2014

This paper proposes a partial answer to the explicit inversion of the tensor tomography problem in two dimensions, by proving injectivity over certain kinds of tensors and providing reconstruction… Expand