We study Hardy spaces $H^p_ u$ of the conjugate Beltrami equation $bar{partial} f= ubar{partial f}$ over Dini-smooth finitely connected domains, for real contractive $ uin W^{1,r}$ with $r>2$, in the range $r/(r-1)<p<infty$. We develop a theory of co
njugate functions and apply it to solve Dirichlet and Neumann problems for the conductivity equation $ abla.(sigma abla u)=0$ where $sigma=(1- u)/(1+ u)$. In particular situations, we also consider some density properties of traces of solutions together with boundary approximation issues.
We consider two inverse problems related to the tokamak textsl{Tore Supra} through the study of the magnetostatic equation for the poloidal flux. The first one deals with the Cauchy issue of recovering in a two dimensional annular domain boundary mag
netic values on the inner boundary, namely the limiter, from available overdetermined data on the outer boundary. Using tools from complex analysis and properties of genereralized Hardy spaces, we establish stability and existence properties. Secondly the inverse problem of recovering the shape of the plasma is addressed thank tools of shape optimization. Again results about existence and optimality are provided. They give rise to a fast algorithm of identification which is applied to several numerical simulations computing good results either for the classical harmonic case or for the data coming from textsl{Tore Supra}.
