Passing from bulk to bulk/surface evolution in the Allen--Cahn equation
- Liero, Matthias
2010 Mathematics Subject Classification
- 35K55 35K20 82C26
- Gradient flow, Allen-Cahn equation, dynamic boundary condition, energy balance, Gamma-convergence
In this paper we formulate a boundary layer approximation for an Allen-Cahn-type equation involving a small parameter $eps$. Here, $eps$ is related to the thickness of the boundary layer and we are interested in the limit when $eps$ tends to 0 in order to derive nontrivial boundary conditions. The evolution of the system is written as an energy balance formulation of the L^2-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 Gamma- and Mosco convergence. By considering different scalings in the boundary layer we obtain different boundary conditions.
- NoDEA Nonlinear Differential Equations Appl., 20 (2013) pp. 919--942.