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

Monotonicity of principal eigenvalue for elliptic operators with incompressible flow: A functional approach