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

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 $Ω\subset \mathbb{R}^N (N\ge 1)$ is bounded with smooth boundary $\partialΩ$, the constants $\varepsilon>0$ and $α>0$ are the diffusion and drift coefficients, respectively, and $m(x)\in C^{2}(\barΩ)$, $V (x)\in C^γ(\barΩ)~(0<γ<1)$ are given functions. For a class of divergence-free drifts where $m$ is a harmonic function in $Ω$ and has no first integral in $H_{0}^{1}(Ω)$, we prove the convergence of the principal eigenpair $(λ_α, ϕ)$ for (0.1) as $α\rightarrow+\infty$, 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 $(λ_α, ϕ)$ for (0.1) as $α\rightarrow+\infty$, which display the visible effects of the large divergence-free drifts on the principal eigenpair $(λ_α, ϕ)$.