Consider the eigenvalue problem of a linear second order elliptic operator: \(\begin{aligned} -D\Delta \varphi -2\alpha \nabla m(x)\cdot \nabla \varphi +V(x)\varphi =\lambda \varphi \ \ \hbox { in }\Omega , \end{aligned}\) complemented by the Dirichlet boundary condition or the following general Robin boundary condition: \( \frac{\partial \varphi }{\partial n}+\beta (x)\varphi =0 \ \ \hbox { on }\partial \Omega , \) where \(\Omega \subset \mathbb {R}^N (N\ge 1)\) is a bounded smooth domain, \(n(x)\) is the unit exterior normal to \(\partial \Omega \) at \(x\in \partial \Omega \) , \(D>0\) and \(\alpha >0\) are, respectively, the diffusion and advection coefficients, \(m\in C^2(\overline{\Omega }),\,V\in C(\overline{\Omega })\) , \(\beta \in C(\partial \Omega )\) are given functions, and \(\beta \) allows to be positive, sign-changing or negative. In this paper, we aim to establish, as \(\alpha \) approaches \(\infty \) , the asymptotic behavior of the principal eigenvalue under appropriate conditions on the advection function \(m\) . For \(N=1\) , we provide a complete characterization of the asymptotic behavior, assuming that the derivative of \(m\) changes sign at most finitely many times. Our findings not only improve upon the previous work in [6, 9, 50], but also partially address some of the open questions posed in [6]. Furthermore, our results elucidate the novel influence of boundary conditions on such asymptotics.
