Cahiers du CEREMADE 

Unité
Mixte de Recherche du C.N.R.S. N°7534 

Abstract : We consider a class of singular perturbation elliptic
boundary value problems depending on a parameter $varepsilon$
which are classical for $varepsilon > 0$ but highly ill  posed
for $varepsilon = 0$ 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
$varepsilon$ decreases, and the limit does not exist in classical
distribution theory (it only may be described in spaces of
analytical functionals not enjoying localization properties). This
phenomenon is associated with the emergence of the new
characteristic parameter $log varepsilon$. 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. 





200611 

22022006 

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