5.12.2020 Uniqueness results for fractional Calder贸n problems (Covi)

The problem of whether one can determine the electrical conductivity inside of a domain by measurements on its boundary is one of the most classical inverse problems for partial differential equations. It was studied by Alberto Calder贸n in the 1940鈥檚 and later published in his seminal paper in the year 1980. In his dissertation, Giovanni Covi at the 蘑菇直播 studies the problem of uniqueness for many nonlocal and fractional Calder贸n problems. Such inverse problems currently constitute a very active and prosperous field of research for the international mathematical community.
Published
5.12.2020

The dissertation of Giovanni Covi studies the fractional Schr枚dinger equation, which appears in the study of anomalous diffusion and serves as a generalization of models currently used in the fields of medical and industrial imaging. The dissertation considers numerous perturbed versions of such equation and establishes uniqueness and recoverability results.

鈥淭he central objective of the field of Inverse problems consists in determining the properties of the interior of an object by studying its exterior, or its surface鈥, says Covi.

鈥淭his kind of problems is of central interest in many practical applications, including modern scanning techniques for medical imaging, mineral prospection, and even space exploration.鈥

One of the main results shows that any local perturbation of the fractional Schr枚dinger equation can be uniquely recovered from the corresponding Dirichlet to Neumann map, even in the higher order case. In particular, unique recoverability and reconstruction was shown to hold for the fractional conductivity equation and the magnetic Schr枚dinger equation. These results promise to have interesting applications in the study of nonlocal phenomena.

Another main result shows that the bundary and the bulk potentials can be simultaneously recovered from boundary data in the case of a local Calderon problem with mixed Robin conditions. The main application of this result arises in the problem of corrosion detection. This result is however also interesting from a theoretical point of view, as such problem is related to the fractional one via the Caffarelli-Silvestre extension.

The dissertation is published in JYU Dissertation series, number 318, 蘑菇直播, Jyv盲skyl盲 2020. 
ISBN 978-951-39-8391-8 (PDF) URN:ISBN:978-951-39-8391-8 ISSN 2489-9003
Link to the publication: 

For further information
Giovanni Covi, giovanni.g.covi@jyu.fi

M.Sc. Giovanni Covi defends his doctoral dissertation in Mathematics "Uniqueness results for fractional Calder贸n problems" at the 蘑菇直播 in 5th of December 2021 at 12 noon. Opponent is University Lecturer Petri Ola from University of Helsinki and Custos is Professor Mikko Salo from 蘑菇直播. The doctoral dissertation is held in English.

The audience can follow the dissertation online.

Link to the online event: 

Phone number to which the audience can present possible additional questions at the end of the event (to the custos): +358 40 8054476