Asymptotics of the principal eigenvalue of a linear elliptic operator with large advection

Consider the eigenvalue problem of a linear second order elliptic operator: \begin{equation} \nonumber -DΔ\varphi -2α\nabla m(x)\cdot \nabla\varphi+V(x)\varphi=λ\varphi\ \ \hbox{ in }Ω, \end{equation} complemented by the Dirichlet boundary condition or the following general Robin boundary condition: $$\frac{\mathrm{\partial }\phi }{\mathrm{\partial }n}+\beta (x)\phi =0\text{ on }\mathrm{\partial }\mathrm{\Omega },$$ where $\mathrm{\Omega }\subset {\mathbb{R}}^N(N\ge 1)$ is a bounded smooth domain, $n(x)$ is the unit exterior normal to $\mathrm{\partial }\mathrm{\Omega }$ at $x\in \mathrm{\partial }\mathrm{\Omega }$, $D>0$ and $\alpha >0$ are, respectively, the diffusion and advection coefficients, $m\in C^2(\bar{\mathrm{\Omega }}),V\in C(\bar{\mathrm{\Omega }})$, $\beta \in C(\mathrm{\partial }\mathrm{\Omega })$ are given functions, and $\beta$ allows to be positive, sign-changing or negative. In \cite{PZZ2019}, the asymptotic behavior of the principal eigenvalue of the above eigenvalue problem as $D\to 0$ or $D\to \mathrm{\infty }$ was studied. In this paper, when $N\ge 2$, under proper conditions on the advection function $m$, we establish the asymptotic behavior of the principal eigenvalue as $\alpha \to \mathrm{\infty }$, and when $N=1$, we obtain a complete characterization for such asymptotic behavior provided $m^{\mathrm{\prime }}$ changes sign at most finitely many times. Our results complement or improve those in \cite{BHN2005,CL2008,PZ2018} and also partially answer some questions raised in \cite{BHN2005}.