Perturbation method for strongly elliptic second order systems with constant coefficients

We consider a classical Dirichlet problem for a strongly elliptic second order system with constant coefficients in Jordan domains in the plane. We show that the solution of the problem can be represented as a functional series in the powers of the parameter governing the deviation of the operator of the system from the Laplacian. This series converges uniformly in the closure of the domain under the assumption that the boundary of the domain and the given boundary function satisfy sufficient regularity conditions: the composition of the boundary function with the trace of a conformal mapping of the unit circle on the domain belongs to the Hölder class with the exponent exceeding 1/2.