On the Robin problem in fractal domains

We study the solution to the Robin boundary problem for the Laplacian in a Euclidean domain. We present some families of fractal domains where the infimum of the solution to the mixed Dirichlet-Robin boundary problem is greater than 0, and some other families of domains where it is equal to 0. We also give a new result on 'trap domains' defined in Burdzy, Chen and Marshall (Math. Z.), that is, domains where reflecting Brownian motion takes a long time to reach the center of the domain.