Criteria for well-posedness of degenerate elliptic and parabolic problems

We provide explicit criteria for uniqueness or nonuniqueness of solutions to a wide class of second order elliptic and parabolic problems. The operator coefficients may be unbounded or vanish, or not to have a limit when approaching some part of the boundary, referred to as singular boundary. We discuss whether boundary conditions should be imposed on such a part to ensure well-posedness. The answer depends on the dimension of the singular boundary, and possibly on the behavior of coefficients near it.