Integral Approach to Sensitive Singular Perturbations

Abstract

The main purpose of this chapter is to give general ideas on a kind of singular perturbation arising 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, as well as an integral heuristic procedure reducing these problems to simpler ones. The system depends essentially on the parameter ε equal to the relative thickness of the shell. It appears that the “limit problem” for ε = 0 is highly ill posed. Indeed, the boundary conditions on the free boundary are not “adapted” to the system of equations; they do not satisfy the Shapiro–Lopatinskii (SL) condition. Roughly speaking, this amounts to some kind of “transparency” of the boundary conditions, which allows some kind of locally indeterminate oscillations along the boundary, exponentially decreasing inside the domain. This pathological behavior only occurs for ε = 0. In fact, for ε > 0 the problem is “classical” When ε is positive but small, the “determinacy” of the oscillations only holds with the help of boundary conditions on other boundaries, as well as the small terms coming from ε > 0.