A variational irreversible thermodynamic method for curved surfaces and interfaces in a two-dimensional (2D) continuum, having anisotropic specific surface Gibbs free energy, is developed by utilizing the more realistic monolayer model of Verschaffelt and Guggenheim for the description of interfaces and surfaces in connection with the global entropy production hypothesis. This approach considers not only the asymmetric disposition of the grain-boundary triple junction, but also its dynamical effects on the morphological evolution of surfaces. The governing Euler equation and the associated boundary conditions (the strong solution) are derived rigorously by the variational technique applied on the positive definite rate of global internal entropy production; the results are in excellent accord with those deduced by the first-principles theory of irreversible thermodynamics of curved surfaces with triple junctions as formulated previously by the author, using the basic postulate of the local internal entropy production in connection with the microfinite-element method in discrete 2D space. At the final stage, the whole problem is converted into a variational extremum problem in order to obtain the weak solution in a class of smooth functions (i.e., Hermite functions) having continuous derivatives C-infinity(-infinity,+infinity) by transforming the displacement field into the particle-flux representation using the principle of conservation of particles, including the phase transition. As an application of the weak solution, which is converted into a compact matrix format in the normalized and scaled time and space domain, a set of computer simulation experiments is performed on symmetrically disposed bicrystal thin metallic films having fourfold anisotropic specific surface Gibbs free energy to demonstrate the breaching effects caused by grain-boundary grooving under the surface drift diffusion driven by the capillarity without electromigration forces.