Estimates in Hölder classes for the solution of an inhomogeneous Dirichlet problem for a singularly perturbed homogeneous convection-diffusion equation

An inhomogeneous Dirichlet boundary value problem for a singularly perturbed homogeneous convection-diffusion equation with constant coefficients is considered in a half-plane. Convection is assumed to be directed orthogonally to the half-plane boundary away from it. Assuming that the boundary function is from the space $C^{2,\lambda}$, $0<\lambda<1$, an unimprovable estimate for the solution bounded at infinity is obtained in the appropriate Hölder norm.