  In this paper we obtain the continuity of attractors for nonlinear parabolic
equations with nonlinear boundary conditions when the boundary of the domain
varies very rapidly as a parameter $\epsilon$ goes to zero. We consider the
case where the boundary of the domain presents a highly oscillatory behavior as
the parameter $\epsilon$ goes to zero. For the case where we have a Lipschitz
deformation of the boundary with the Lipschitz constant uniformly bounded in
$\epsilon$ but the boundaries do not approach in a Lipschitz sense, the
solutions of these equations converge in certain sense to the solution of a
limit parabolic equation of the same type but where the boundary condition has
a factor that captures the oscillations of the boundary. To address this
problem, it is necessary to consider the notion of convergence of functions
defined in varying domains and the convergence of a family of operators defined
in different Banach spaces. Moreover, since we consider problems with nonlinear
boundary conditions, it is necessary to extend these concepts to the case of
spaces with negative exponents and to operators defined between these spaces.
