On a coupled bulk–surface Allen–Cahn system with an affine linear transmission condition and its approximation by a Robin boundary condition

We study a coupled bulk–surface Allen–Cahn system with an affine linear transmission condition, that is, the trace values of the bulk variable and the values of the surface variable are connected via an affine relation, and this serves to generalize the usual dynamic boundary conditions. We tackle the problem of well-posedness via a penalization method using Robin boundary conditions. In particular, for the relaxation problem, the strong well-posedness and long-time behaviour of solutions can be shown for more general and possibly nonlinear relations. New difficulties arise since the surface variable is no longer the trace of the bulk variable, and uniform estimates in the relaxation parameter are scarce. Nevertheless, weak convergence to the original problem with affine linear relations can be shown. Using the approach of Colli and Fukao (2015), we show strong existence to the original problem with affine linear relations, and derive an error estimate between solutions to the relaxed and original problems.