  We consider the small mass asymptotic (Smoluchowski-Kramers approximation)
for the Langevin equation with a variable friction coefficient. The friction
coefficient is assumed to be vanishing within certain region. We introduce a
regularization for this problem and study the limiting motion for the
1-dimensional case and a multidimensional model problem. The limiting motion is
a Markov process on a projected space. We specify the generator and boundary
condition of this limiting Markov process and prove the convergence.
