SENSITIVE VERSUS CLASSICAL SINGULAR PERTURBATION PROBLEM VIA FOURIER TRANSFORM

Abstract

We consider a class of singular perturbation 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. This kind of problems is motivated by certain situations in thin shell theory, but we only deal here with model problems and geometries allowing a Fourier transform treatment. We consider more general loadings and more singular perturbation terms than in previous works on the subject. The asymptotic process exhibits a complexification phenomenon: in some sense, the solution becomes more and more complicated as ε decreases, and the limit does not exist in classical distribution theory (it may only be described in spaces of analytical functionals not enjoying localization properties). This phenomenon is associated with the emergence of the new characteristic parameter |log ε|. Numerical experiments based on a formal asymptotics are presented, exhibiting features which are unusual in classical elliptic equations theory. We also give a Fourier transform treatment of classical singular perturbations in order to exhibit the drastic differences with the present situation.