Abstract
In this paper we formulate a boundary layer approximation for an Allen–Cahn-type equation involving a small parameter \({\varepsilon}\) . Here, \({\varepsilon}\) is related to the thickness of the boundary layer and we are interested in the limit \({\varepsilon \to 0}\) in order to derive nontrivial boundary conditions. The evolution of the system is written as an energy balance formulation of the L2-gradient flow with the corresponding Allen–Cahn energy functional. By transforming the boundary layer to a fixed domain we show the convergence of the solutions to a solution of a limit system. This is done by using concepts related to Γ- and Mosco convergence. By considering different scalings in the boundary layer we obtain different boundary conditions.
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