Singular perturbation and interpolation

It is well known that the rate of convergence of the solution u(epsilon) of a singular perturbed problem to the solution u of the unperturbed equation can be measured in terms of the ''smoothness'' of u; smoothness which, in turn, can be expressed in terms of linear interpolation theory. We want to prove a closer relationship between interpolation and singular perturbations, showing that interpolate spaces can be characterized by such a rate of convergence. Furthermore, with respect to a suitable (quite natural) definition of interpolation between convex sets, such a characterization holds true also in the framework of variational inequalities.