  In this paper, we develop a viscosity method for Homogenization of Nonlinear
Parabolic Equations constrained by highly oscillating obstacles or Dirichlet
data in perforated domains. The Dirichlet data on the perforated domain can be
considered as a constraint or an obstacle. Homogenization of nonlinear eigen
value problems has been also considered to control the degeneracy of the Porous
medium equation in perforated domains. For the simplicity, we consider
obstacles that consist of cylindrical columns distributed periodically and
perforated domains with punctured balls. If the decay rate of the capac- ity of
columns or the capacity of punctured ball is too high or too small, the limit
of u\k{o} will converge to trivial solutions. The critical decay rates of
having nontrivial solution are obtained with the construction of barriers. We
also show the limit of u\k{o} satisfies a homogenized equation with a term
showing the effect of the highly oscillating obstacles or perforated domain in
viscosity sense.
