The boundary perturbation method combined with the superposition principle is used to calculate the stress concentration along the arbitrary curved interface of an isotropic thin film coherently bonded to a substrate. In the case of plane strain conditions, the boundary value problem is formulated for a four-phase system involving two-dimensional constitutive equations for bulk materials and one-dimensional equations of Gurtin–Murdoch model for surface and interface. Static boundary conditions are formulated in the form of generalized Young–Laplace equations. Kinematic boundary conditions describe the continuous of displacements across the surface and interphase regions. Using Goursat–Kolosov complex potentials, the system of boundary equations is reduced to a system of the integral equations via first-order boundary perturbation method. Finally, the solution of boundary value problem is obtained in terms of Fourier series. The numerical analysis is then carried out using the practically important properties of ultra-thin-film materials.