ON NONUNIQUENESS FOR THE ANISOTROPIC CALDERÓN PROBLEM WITH PARTIAL DATA
| Autor: | Niky Kamran, François Nicoleau, Thierry Daudé |
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| Rok vydání: | 2020 |
| Předmět: |
Statistics and Probability
Pure mathematics Algebra and Number Theory 010102 general mathematics Hölder condition Order (ring theory) Boundary (topology) Conformal map Harmonic (mathematics) 01 natural sciences Manifold Theoretical Computer Science Computational Mathematics Product (mathematics) 0103 physical sciences Metric (mathematics) Discrete Mathematics and Combinatorics 010307 mathematical physics Geometry and Topology 0101 mathematics Mathematical Physics Analysis Mathematics |
| Zdroj: | Forum of Mathematics, Sigma. 8 |
| ISSN: | 2050-5094 |
| DOI: | 10.1017/fms.2020.1 |
| Popis: | 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$\unicode[STIX]{x1D70C}at the end where the measurements are made. More precisely, we construct a toroidal ring$(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. |
| Databáze: | OpenAIRE |
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