Thesis topic: L^2-Theory for nonlocal operators on domains.

FOGHEM GOUNOUE Guy Fabrice, AIMS-Cameroon alumnus of the 2015/2016 batch has successfully earned a PhD in Mathematics, precisely in the field of Analysis of Nonlocal Operators from the University of Bielefeld, Germany.

Realized under the supervision of Prof. Dr. Moritz Kassmann, his overreaching goal was to study the well-posedness of a certain class of ellipics IntegroDifferential Equations (IDEs) and their applications within the frameworks of Hilbert spaces, and to explore the asymptotic behaviour of solutions associated with integrodifferential operators with collapsing jump kernels.

The latter objective aims to bridge a transition from nonlocal concepts to the corresponding local concepts. For instance, while conciliating nonlocal equations and local equations, we show that limits of solutions to elliptic IDEs are solutions to elliptic partial differential equations (PDEs) of second order”, he elucidated during his virtual defense on September 14, 2020.
A part of this interesting finding has been published in the Nonlinear Analysis journal under the title “Mosco Convergence of nonlocal to local quadratic forms”.

Raising the natural conceptual problem regarding the function spaces related to elliptic IntegroDifferential equations (IDEs), Fabrice’s work further sets some milestones for a method of dealing with the IDEs complement value problems in the framework of Hilbert spaces.

The applicability of his work can be visible in areas including the thin obstacle problem, optimization, finance, phase transitions, ratified materials, anomalous diffusion, crystal dislocation, soft thin films, in some models of semipermeable membranes and flame propagation, in conservation laws, in the ultra-relativistic limit of quantum mechanics, in quasi-geostrophic flows, in materials science, and in water waves.

Dr. Foghem is currently completing his term as Teaching Assistant at Bielefeld University while envisaging a Postdoc position at the University of Dresden, Germany. His research interests are in Ingegro-Differential Equations (IDEs) and their applications.