Analytical methods for a selection of elliptic singular perturbation problems
We consider several model problems from a class of elliptic perturbation equations in two dimensions. The domains, the differential operators, the boundary conditions, and so on, are rather simple, and are chosen in a way that the solutions can be obtained in the form of integrals or Fourier series. By using several techniques from asymptotic analysis (saddle point methods, for instance) we try to construct asymptotic approximations with respect to the small parameter that multiplies the differential operator of highest order. In particular we consider approximations that hold uniformly in the so-called boundary layers. We also pay attention to how to obtain a few terms in the asymptotic expansion by using direct methods based on singular perturbation methods.