Analytic singular perturbations of elliptic systems

We study a singular perturbation problem for a system defined under a variational form. We show the analytic dependence of the solution of the equation with respect to a small, nonnull parameter ε, and make explicit the terms of the power series. This result improves a theorem of Chap. I of J. L. Lions (“Perturbations singulières dans les problèmes aux limites et en contrôle optimal,” Springer-Verlag, Berlin 1973) in which the variational forms are supposed to be symmetric and no analycity result is given. We give an application to the study of a stationary thermical system with a small convection coefficient.