Rigorous and heuristic treatment of certain sensitive singular perturbations

Abstract

We consider singular perturbated elliptic boundary value problems depending on a parameter ε which are classical for ε > 0 but highly ill-posed for ε = 0 as the boundary condition does not satisfy the Shapiro–Lopatinskii condition on a part of the boundary. We mainly use a limit device due to Caillerie, using a non-variational framework which proves the existence of a unique limit in an appropriate abstract space. We consider more general domains (two dimensional manifold with boundary) than in previous works on the subject. To do so, we use a heuristic reasoning allowing some simplifications which show the “equivalence” of the problem in the general geometry with another one in a one dimensional manifold. This kind of problems is motivated by certain situations in thin shell theory.