Exponential decay phenomenon of the principal eigenvalue of an elliptic   operator with a large drift term of gradient type
  We study an asymptotic behaviour of the principal eigenvalue for an elliptic operator with large advection which is given by a gradient of a potential function. It is shown that the principal eigenvalue decays exponentially under the velocity potential well condition as the parameter tends to infinity. 
Shuichi Jimbo, Masato Kimura and Hirofumi Notsu