Section: Application Domains

Free boundary problems

Participants : Laurent Baratchart, Juliette Leblond, Slah Chaabi.

The team has engaged in the study of problems with variable conductivity $\sigma$, governed by a 2-D equation of the form $\text{div}\left(\sigma \nabla u\right)=0$. Such equations are in one-to-one correspondence with real parts of solutions to conjugate-Beltrami equations $\overline{\partial }f=\nu \overline{\partial f}$, so that complex analysis is a tool to study them, see [4] , [13] , [28] . This research was prompted by issues in plasma confinement for thermonuclear fusion in a tokamak, more precisely with the extrapolation of magnetic data on the boundary of the chamber from the outer boundary of the plasma, which is a level curve for the poloidal flux solving the original div-grad equation. Solving this inverse problem of Bernoulli type is of importance to determine the appropriate boundary conditions to be applied to the chamber in order to shape the plasma [58] . This research was started in collaboration with CEA-IRFM (Cadarache) and the Laboratoire J.-A. Dieudonné at the Univ. of Nice-SA. Within the team, it is now expanding to cover Dirichlet-Neumann problems for larger classes of conductivities, cf. in particular, the PhD thesis of S. Chaabi [12] , [28] , jointly supervised with the CMI-LATP at the Aix-Marseille University. (see Section  6.2 ).