scholarly journals Unique Continuation, Runge Approximation and the Fractional Calderón Problem

2018 ◽  
pp. 1-10 ◽  
Author(s):  
Angkana Rüland
Keyword(s):  
Unique Continuation ◽  
Calderón Problem

scholarly journals ON NONUNIQUENESS FOR THE ANISOTROPIC CALDERÓN PROBLEM WITH PARTIAL DATA

2020 ◽  
Vol 8 ◽  
Author(s):  
THIERRY DAUDÉ ◽  
NIKY KAMRAN ◽  
FRANÇOIS NICOLEAU
Keyword(s):  
Infinite Number ◽  
Riemannian Manifolds ◽  
Riemannian Metrics ◽  
Unique Continuation ◽  
Manifolds With Boundary ◽  
Conformal Class ◽  
Hölder Continuous ◽  
Continuation Principle ◽  
Partial Data ◽  
Calderón Problem

We show that there is nonuniqueness for the Calderón problem with partial data for Riemannian metrics with Hölder continuous coefficients in dimension greater than or equal to three. We provide simple counterexamples in the case of cylindrical Riemannian manifolds with boundary having two ends. The coefficients of these metrics are smooth in the interior of the manifold and are only Hölder continuous of order $\unicode[STIX]{x1D70C}<1$ at the end where the measurements are made. More precisely, we construct a toroidal ring $(M,g)$ and we show that there exist in the conformal class of $g$ an infinite number of Riemannian metrics $\tilde{g}=c^{4}g$ such that their corresponding partial Dirichlet-to-Neumann maps at one end coincide. The corresponding smooth conformal factors are harmonic with respect to the metric $g$ and do not satisfy the unique continuation principle.

scholarly journals Runge approximation and stability improvement for a partial data Calderón problem for the acoustic Helmholtz equation

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
María Ángeles García-Ferrero ◽  
Angkana Rüland ◽  
Wiktoria Zatoń
Keyword(s):  
Helmholtz Equation ◽  
Frequency Dependence ◽  
High Frequency ◽  
Convex Sets ◽  
Unique Continuation ◽  
Approximation Properties ◽  
Frequency Limit ◽  
Partial Data ◽  
Calderón Problem ◽  
Quantitative Unique Continuation

<p style='text-indent:20px;'>In this article, we discuss quantitative Runge approximation properties for the acoustic Helmholtz equation and prove stability improvement results in the high frequency limit for an associated partial data inverse problem modelled on [<xref ref-type="bibr" rid="b3">3</xref>,<xref ref-type="bibr" rid="b35">35</xref>]. The results rely on quantitative unique continuation estimates in suitable function spaces with explicit frequency dependence. We contrast the frequency dependence of interior Runge approximation results from non-convex and convex sets.</p>

scholarly journals Quantitative Runge Approximation and Inverse Problems

2018 ◽  
Vol 2019 (20) ◽  
pp. 6216-6234 ◽  
Author(s):  
Angkana Rüland ◽  
Mikko Salo
Keyword(s):  
Inverse Problems ◽  
Approximation Property ◽  
Short Note ◽  
Unique Continuation ◽  
Local Data ◽  
Model Application ◽  
Quantitative Version ◽  
Calderón Problem ◽  
Quantitative Unique Continuation ◽  
Second Order Elliptic Operators

AbstractIn this short note, we provide a quantitative version of the classical Runge approximation property for second-order elliptic operators. This relies on quantitative unique continuation results and duality arguments. We show that these estimates are essentially optimal. As a model application, we provide a new proof of the result from [8], [2] on stability for the Calderón problem with local data.

Uniqueness in the Calderón problem and bilinear restriction estimates

2021 ◽  
pp. 109119
Author(s):  
Seheon Ham ◽  
Yehyun Kwon ◽  
Sanghyuk Lee
Keyword(s):  
Calderón Problem

scholarly journals Calderón problem for Yang–Mills connections

2020 ◽  
Vol 10 (2) ◽  
pp. 463-513
Author(s):  
Mihajlo Cekić
Keyword(s):  
Yang Mills ◽  
Calderón Problem

scholarly journals Observability and unique continuation inequalities for the Schrödinger equation

2019 ◽  
Vol 21 (11) ◽  
pp. 3513-3572
Author(s):  
Gengsheng Wang ◽  
Ming Wang ◽  
Yubiao Zhang
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Unique Continuation

scholarly journals Strict monotonicity and unique continuation of the biharmonic operator - doi: 10.5269/bspm.v30i2.14502

1970 ◽  
Vol 30 (2) ◽  
pp. 79-83
Author(s):  
Najib Tsouli ◽  
Omar Chakrone ◽  
Mostafa Rahmani ◽  
Omar Darhouche
Keyword(s):  
Unique Continuation ◽  
Biharmonic Operator ◽  
Strict Monotonicity ◽  
Unique Continuation Property ◽  
Continuation Property

In this paper, we will show that the strict monotonicity of the eigenvalues of the biharmonic operator holds if and only if some unique continuation property is satisfied by the corresponding eigenfunctions.

scholarly journals Calderón problem for Maxwell's equations in two dimensions

2016 ◽  
Vol 24 (3) ◽  
Author(s):  
Oleg Y. Imanuvilov ◽  
Masahiro Yamamoto
Keyword(s):  
Maxwell’S Equations ◽  
Maxwell Equations ◽  
Maxwell's Equations ◽  
Two Dimensions ◽  
Two Dimensional ◽  
Global Uniqueness ◽  
Calderón Problem ◽  
Dirichlet To Neumann Map

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

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