We consider the small mass asymptotics (Smoluchowski-Kramers approximation) for the Langevin equation with a variable friction coefficient. The limit of the solution in the classical sense does not exist in this case. We study a modification of the Smoluchowski-Kramers approximation. Some applications of the Smoluchowski-Kramers approximation to problems with fast oscillating or discontinuous coefficients are considered. Bibliography: 15 titles.
Bibliographical noteFunding Information:
This work is supported in part by NSF Grants DMS-0803287 and DMS-0854982.