Integral approach to sensitive singular perturbations

We consider singular perturbation elliptic problems depending on a parameter ε such that, for ε = 0 the boundary conditions are not adapted to the equation (they do not satisfy the Shapiro Lopatinskii condition). The limit only holds in very abstract spaces out of distribution theory involving complexification and non-local phenomena. We give a very elementary model problem showing the main features of the limit process, as well as a heuristic integral procedure for obtaining a description of the solutions for small ε. Such kind of problems appear in thin shell theory when the middle surface is elliptic and the shell is fixed by a part of the boundary and free by the rest.