We present a constructive method for the robust approximation to solutions of some elliptic equations in a plane domain from incomplete and corrupted boundary data. We state this inverse problem in generalized Hardy spaces of functions satisfying the conjugate Beltrami equation, of which we give some properties, in the Hilbertian framework. The issue is then reworded as a constrained approximation (bounded extremal) problem which is shown to be well-posed. A practical motivation comes from modelling plasma confinement in a tokamak reactor. There, the particular form of the conductivity coefficient leads to Bessel-exponential type families of solutions of which we establish density properties.