Asymptotic Behavior of the Principal Eigenvalue Problems with Large Divergence-Free Drifts

Abstract

In this paper, we consider the following principal eigenvalue problem with a large divergence-free drift: \begin{equation}\label{0.1} -\varepsilonΔϕ-2α\nabla m(x)\cdot\nabla ϕ+V(x)ϕ=λ_αϕ \,\ \text{in}\, \ H_0^1(Ω),\tag{0.1} \end{equation} where the domain $\hat{I}\text{c◯}\subset {\mathbb{R}}^N(N\ge 1)$ is bounded with smooth boundary $\partial \hat{I}\text{c◯}$, the constants $\epsilon >0$ and $\hat{I}\pm >0$ are the diffusion and drift coefficients, respectively, and $m(x)\in C^2(\bar{\hat{I}}\text{c◯})$, $V(x)\in C^3(\bar{\hat{I}}\text{c◯})(0<{\hat{I}}^3<1)$ are given functions. For a class of divergence-free drifts where $m$ is a harmonic function in $\hat{I}\text{c◯}$ and has no first integral in $H_0^1(\hat{I}\text{c◯})$, we prove the convergence of the principal eigenpair $(\hat{I}{»}_{\hat{I}}\pm ,\ddot{I}•)$ for (0.1) as $\hat{I}\pm \to +\infty$, which addresses a special case of the open question proposed in [H. Berestycki, F. Hamel and N. Nadirashvili, CMP, 2005]. Moreover, we further investigate the refined limiting profiles of the principal eigenpair $(\hat{I}{»}_{\hat{I}}\pm ,\ddot{I}•)$ for (0.1) as $\hat{I}\pm \to +\infty$, which display the visible effects of the large divergence-free drifts on the principal eigenpair $(\hat{I}{»}_{\hat{I}}\pm ,\ddot{I}•)$.

31 pages