  We establish the monotonicity of the principal eigenvalue $\lambda_1(A)$, as
a function of the advection amplitude $A$, for the elliptic operator
$L_{A}=-\mathrm{div}(a(x)\nabla)+A\mathbf{V}\cdot\nabla +c(x)$ with
incompressible flow $\mathbf{V}$, subject to Dirichlet, Robin and Neumann
boundary conditions. As a consequence, the limit of $\lambda_1(A)$ as $A\to
\infty$ always exists and is finite for Robin boundary conditions. These
results answer some open questions raised by [Berestycki, H., Hamel, F.,
Nadirashvili, N.: Elliptic eigenvalue problems with large drift and
applications to nonlinear propagation phenomena, Commun. Math. Phys. 253,
451-480 (2005)]. Our method relies upon some functional which is associated
with principal eigenfuntions for operator $L_A$ and its adjoint operator. As a
byproduct of the approach, a new min-max characterization of $\lambda_1(A)$ is
given.
