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

TL;DR

This paper proves that the principal eigenvalue of certain elliptic operators with incompressible flow increases monotonically with advection amplitude, establishing limits and providing a new characterization method.

Contribution

It introduces a functional approach to prove monotonicity and limits of the principal eigenvalue, answering open questions in the field.

Findings

Principal eigenvalue $\lambda_1(A)$ is monotonic with respect to advection amplitude $A$.

Limit of $\lambda_1(A)$ exists and is finite for Robin boundary conditions as $A o \infty$.

A new min-max characterization of $\lambda_1(A)$ is established.

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.