Rigorous and Heuristic Treatment of Sensitive Singular Perturbations Arising in Elliptic Shells

Abstract

We consider singular perturbations of elliptic systems 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 holds only in very abstract spaces out of distribution theory involving complexification and nonlocal phenomena. This system appears in thin shell theory when the middle surface is elliptic and the shell is fixed on a part of the boundary and free on the rest. We use a heuristic reasoning applying some simplifications which allow us to reduce the original problem in a domain to another problem on its boundary. The novelty of this work is that we consider systems of partial diferential equations while in our previous work we were dealing with single equations.