Asymptotic expansion of the solution of a boundary value problem

Abstract

We study, in the rectangle Ω=(0,a)× (0,b), the Dirichlet boundary value problem for the elliptic partial differential equation $$Lu = - \varepsilon \Delta u + pu_x + gu_y + qu = f$$ , where 0 0,q≤0. We construct a formal asymptotic expansion of the solutionu of this problem for small e. This expansion contains the solution of the reduced equation and boundary layer functions. The parabolic boundary layer functions satisfy a parabolic equation with an unbounded coefficient. We transform the parabolic equation into a heat equation to develop properties of the parabolic boundary layer. Estimates for the remainder in the expansion are established that are of the order of magnitude of powers of e.