WIAS Preprint No. 1676, (2012)

Passing from bulk to bulk/surface evolution in the Allen--Cahn equation



Authors

  • Liero, Matthias
    ORCID: 0000-0002-0963-2915

2010 Mathematics Subject Classification

  • 35K55 35K20 82C26

Keywords

  • Gradient flow, Allen-Cahn equation, dynamic boundary condition, energy balance, Gamma-convergence

Abstract

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.

Appeared in

  • NoDEA Nonlinear Differential Equations Appl., 20 (2013) pp. 919--942.

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