Uniqueness results for fractional Calderón problems
This dissertation studies the inverse problem for a specific partial differential equation,
the so called fractional Calderón problem or inverse problem for the fractional Schrödinger
equation. The dissertation focuses mainly on uniqueness results for inverse problems
involving the Dirichlet to Neumann map, the object encoding exterior measurements in
the model. The included articles show how this information suffices to determine the
parameters involved in the problems considered.
The first article considers a fractional version of the inverse problem for the conductivity
equation, showing that the unknown conductivity can be recovered from the DN map even
in the case of a single measurement. The technique employed is the fractional Liouville
reduction, which allows one to state the problem in terms of the fractional Schrödinger
equation. The second article extends the known result for the fractional Schrödinger
equation to the magnetic case, showing how a nonlocal perturbation and a potential can
be both recovered up to a natural gauge. This resembles the results known for the local
case. The third article explores the fractional Schrödinger equation in a high order regime,
proving the injectivity of the relative DN map in both the perturbed and unperturbed
cases. This requires a high order Poincaré inequality, which has been studied in the
same paper. The fifth article follows the third one, extending the study to general local
highorder perturbations: the coefficients of any local lower order operator are shown to
be recoverable from the DN map. The fourth article studies the perturbed fractional
Calderón problem by means of the CaffarelliSilvestre extension, transforming it into a
local problem with mixed Robin boundary conditions, eventually showing that the bulk
and boundary potentials can be recovered simultaneously. This requires some technical
Carleman estimates and the construction of a new class of CGO solutions.
The introduction of the dissertation contains a survey of the literature related to both
the classical and fractional Calderón problems, as well as a collection of the definitions of
the function spaces appearing in the articles. The appendix is an informal introduction
to key concepts in inverse problems and EIT, thought for the use of the general public.
...
Publisher
Jyväskylän yliopistoISBN
9789513983918ISSN Search the Publication Forum
24899003Contains publications
 Artikkeli I: Covi, G. (2020). Inverse problems for a fractional conductivity equation. Nonlinear Analysis: Theory, Methods and Applications, 193, 111418. DOI: 10.1016/j.na.2019.01.008
 Artikkeli II: Covi, Giovanni (2020). An inverse problem for the fractional Schrödinger equation in a magnetic ﬁeld. Inverse Problems, 36 (4), 045004. DOI: 10.1088/13616420/ab661a
 Artikkeli III: Covi, Giovanni; Mönkkönen, Keijo and Railo, Jesse (2020) Unique continuation property and Poincaré inequality for higher order fractional Laplacians with applications in inverse problems. Preprint arXiv: 2001.06210v2
 Artikkeli IV: Covi, Giovanni and Rüland, Angkana (2020) On some partial data Calderón type problems with mixed boundary conditions. Preprint. arXiv: 2006.03252v2
 Artikkeli V: Covi, Giovanni; Mönkkönen, Keijo; Railo, Jesse and Uhlmann, Gunther (2020) The higher order fractional Calderón problem for linear local operators: uniqueness. Preprint. arXiv: 2008.10227
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