Asymptotic solution of the Dirichlet problem for a ring, when the corresponding unperturbed equation has a regular special circle

The article investigates the Dirichlet problem in a ring for a linear inhomogeneous second-order elliptic equation with two independent variables. In the equation under consideration, there is a small parameter at the highest derivatives, i.e. at the Laplacian. A solution to the Dirichlet problem for a ring, based on the theory of partial differential equations, exists and is unique. However, attempts to construct an explicit solution to the Dirichlet problem and to determine the dependence of the solution on a small parameter directly failed. It is required to construct a complete uniform asymptotic expansion of the solution of the Dirichlet problem for a ring in powers of a small parameter. The problem under consideration has two features: the first one is a small parameter at the Laplacian and the second one is that the corresponding unperturbed equation has a regular special line. This line is a circle. Therefore, when constructing an asymptotic solution, there appear additional difficulties. The formal asymptotic solution of the Dirichlet problem for a ring is constructed by the generalized method of boundary functions. Using the maximum principle, the constructed formal asymptotic solution is substantiated. The constructed decomposition is asymptotic in the sense of Erdélyi.
The results obtained can find applications in continuum mechanics, hydro- and aerodynamics, magneto hydrodynamics, oceanology, etc.