I'm not in the field, but some of your questions are research problems regarding boundary conditions of the laplacian operator. There's also similar work frame

I'm not in the field, but some of your questions are research problems regarding boundary conditions of the laplacian operator. There's also similar work framed in terms of billiards: https://en.m.wikipedia.org/wiki/Dynamical_billiards https://en.m.wikipedia.org/wiki/Dynamical_billiards