Asymptotic and spectral properties of a class of singular-stiff problems.(English)Zbl 0833.47011

Summary: We consider a somewhat general framework for perturbation problems involving a small parameter $$\varepsilon \searrow 0$$. This framework contains as particular cases singular perturbations, stiff problems and elastic shell theory. Different results appear according to the nature of the limit problem $$\varepsilon =0$$: inhibited (mainly singular perturbations) and non-inhibited (mainly stiff problems). Somewhat elementary results are given for the static problem in the non-inhibited case and for evolution problems in time. The spectral properties are then considered in different ways according to the compactness hypotheses and to the range of frequencies (low or medium). In the case when the limit problem is not compact, we use a Fourier transform technique from time to the spectral parameter which provide convergence results for the spectral families. An essential spectrum is involved in general. Then we consider the static problem in the inhibited case (this is, in particular, the problem of the justification of the membrane approximation for shells which do not admit pure flexions). The convergence $$\varepsilon \searrow 0$$ is then proved for forcing terms $$f$$ belonging to the standard space $$H$$ (resp. to a dense set of the standard space $$H$$) when the origin does not belong (resp. belongs) to the essential spectrum of the limit problem. An appendix contains applications to elastic thin shells.

MSC:

 47A55 Perturbation theory of linear operators 35P05 General topics in linear spectral theory for PDEs 74K15 Membranes