Approximation of dynamic boundary condition: The Allen--Cahn equation

The talk gives insight into the ongoing work on the speaker's PhD thesis, which deals with dynamic interfaces in the semiconductor theory. By means of the Allen--Cahn equation - as a simple toy model - it is shown how dynamic boundary conditions, like the one treated by Sprekels and Wu in [2], can be obtained by considering a system of parabolic equations in the bulk and in a boundary layer whose thickness tends to zero. Different scalings of the coefficients give rise to different dynamics on the boundary. The approach is based on the gradient flow structure of the Allen--Cahn equation involving energy and dissipation functionals. 

Concludingly, a new way to look at semiconductor equations in the framework of generalized gradient flows introduced by Mielke in [1] is given. Here, the metric is constructed by using the dual dissipation potential, which is a convex function of the chemical potentials. In particular, it is possible to treat diffusion and reaction terms simultaneously.

[1] Mielke, Alexander: A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems, to appear.

[2] Sprekels, Jurgen, Wu, Hao: A note on a parabolic equation with nonlinear dynamical boundary condition, Nonlinear Anal., 72 (2010) pp. 3028--3048.