Stabilité et convergence dans les problèmes de perturbation singulière

Abstract

It is shown that Sobolev spaces with two indices H^s_ε(X), s=(s₁,s₂)∈R² are proper approximations of usual Sobolev spaces H^S₁(X). It is also stated that, under natural conditions, an estimate of the form ‖u‖_Hε≤C{‖A_εu‖_Fε+‖u‖_Eε} for an operator A_ε from a space H_ε to a space F_ε can be sharpened and can lead to the inverse stability of A_ε. Under minimal assumptions on the data, these results can be used for establishing convergence results for solutions of classical or Wiener–Hopf coercive singularly perturbed problems investigated by Frank and Wendt [10,11,12,37,38] and for solutions of boundary value problems in L_p-spaces.