Asymptotics of the solution to the mixed boundary elliptic problem

The following two-dimension problem is considered: equation Δu = f(x) in some domain Ω ∈ ℝ² with piecewise smooth boundary. The boundary condition is following: the derivative on a normal is equal zero everywhere, except a small segment γ, where function u(x) is given. The length of the segment equal to a small parameter ε. There is a problem to find the asymptotics of the solution u(x, ε) as ε → 0. The full asymptotic expansion was constructed and proved.