Elliptic eigenvalue problems with large drift and applications to nonlinear propagation phenomena

This paper is concerned with the asymptotic behaviour of the principal eigenvalue of some linear elliptic equations in the limit of high first-order coefficients. Roughly speaking, one of the main results says that the principal eigenvalue, with Dirichlet boundary conditions, is bounded as the amplitude of the coefficients of the first-order derivatives goes to infinity if and only if the associated dynamical system has a first integral, and the limiting eigenvalue is then determined through the minimization of the Dirichlet functional over all first integrals. A parabolic version of these results, as well as other results for more general equations, are given. Some of the main consequences concern the influence of high advection or drift on the speed of propagation of pulsating travelling fronts.