CERAMATHS - DMATHS seminar: presentation by Faker Ben Belgacem

The CERAMATHS mathematics department seminar will welcome Faker Ben Belgacem (UT Compiègne) at 2pm on Thursday, October 10, 2024, for the following talk:

Singularities and treatment for elliptic problems with variable coefficients and Dirac sources

The aim is to study the structure of potentials generated by point Dirac sources in hybrid or composite conductive media, and which are solutions of diffusion problems.
The complication arises from the following facts:
(i.)- the conductivity of the medium is variable
. (ii.)- it can jump
. (iii.)- Dirac sources are located at the interfaces of discontinuities
. This model is used in many fields. The most emblematic for the team is that of epileptic sources in the cerebral cortex.
The potential created lacks regularity (its energy is infinite) and standard variational formulations no longer apply. The adapted variational problem is that used by G. Stampachia; it is written in L2 space and arises from a duality argument. That it admits a unique solution follows from a very fine result of elliptic regularity established by E. Di-Giorgi [Mem. Accad. Sci. Torino, 3, 1957]. The first step (work carried out by F. Ben Belgacem and E. Bejaoui) is a key decomposition in which the singular behavior of the potential is explicitly detected and exhibited.
This singular contribution is given by an equation in which the potential's singular behavior is explicitly detected and exhibited. This singular contribution is given by a mathematical formula while the residual correction is regular and can be simulated by standard variational numerical methods (written in H1).
The talk concludes with a brief overview of current achievements (in collaboration with D. Brancherie) for linear elasticity problems with variable Lamé parameters.