Asymptotic solution of the perturbed first boundary value problem with a non-smooth coefficient

In this paper, we consider the first boundary value problem, that is the Dirichlet problem in a ring for a linear inhomogeneous second-order elliptic equation with two independent variables containing a small parameter in front of the Laplacian. The equation potential is not a smooth function in the field under study. There exists a unique solution of the first boundary value problem under consideration. It is impossible to construct an obvious solution of the first boundary value problem. We are interested in the influence of the small parameter on the solution of the Dirichlet problem in the field under study when the small parameter tends to zero. That is why we need to construct an asymptotic solution of the first boundary value problem in a ring. The problem under consideration has two singularities (a bisingular problem): presence of a small parameter in front of the Laplacian, and solution of a relevant unperturbed equation is not a smooth function in the field under study. To construct an asymptotic solution, we use a modified method of boundary functions since it is impossible to use a classical method of boundary functions. To begin with, we construct a formal asymptotic solution as per the small parameter, and then we evaluate the remainder term of the asymptotic expansion. As a result, we have constructed complete uniform asymptotic expansion of the first boundary value problem in a ring as per the small parameter. The constructed series of the solution of the first boundary value problem is asymptotic in the sense of Erdey.