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

Abstract

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.

13 pages, 0 figures