This work gives an overview of the solution to linear forward and inverse problems for a class of elliptic partial differential equations in two-dimensional domains Ω, in the framework of Banach spaces and operators. We focus on the equation ∇(σ∇u) = 0, with σ in Sobolev spaces W 1,p (Ω), 1 < p < ∞, with Dirichlet or Neumann or mixed boundary conditions. For domains Ω with Dini-smooth boundary ∂Ω, the approach is based on the properties of generalized Hardy spaces.