A functional approach towards eigenvalue problems associated with incompressible flow

We propose a certain functional which is associated with principal eigenfunctions of the elliptic operator L_A = −div(a(x)∇) + AV · ∇ + c(x) and its adjoint operator for general incompressible flow V. The functional can be applied to establish the monotonicity of the principal eigenvalue λ₁(A), as a function of the advection amplitude A, for the operator L_A subject to Dirichlet, Robin and Neumann boundary conditions. This gives a new proof of a conjecture raised by Berestycki, Hamel and Nadirashvili [5]. The functional can also be used to prove the monotonicity of the normalized speed c^∗(A)/A for general incompressible flow, where c^∗(A) is the minimal speed of traveling fronts. This extends an earlier result of Berestycki [3] for steady shear flow.