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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2303.16399 |
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Table of Contents:
- Consider the eigenvalue problem of a linear second order elliptic operator: \begin{equation} \nonumber -DΔφ-2α\nabla m(x)\cdot \nablaφ+V(x)φ=λφ \ \hbox{ in }Ω, \end{equation} complemented by the Dirichlet boundary condition or the following general Robin boundary condition: $$ \frac{\partialφ}{\partial n}+β(x)φ=0 \ \ \hbox{ on }\partialΩ, $$ where $Ω\subset\mathbb{R}^N (N\geq1)$ is a bounded smooth domain, $n(x)$ is the unit exterior normal to $\partialΩ$ at $x\in\partialΩ$, $D>0$ and $α>0$ are, respectively, the diffusion and advection coefficients, $m\in C^2(\overlineΩ),\,V\in C(\overlineΩ)$, $β\in C(\partialΩ)$ are given functions, and $β$ allows to be positive, sign-changing or negative. In \cite{PZZ2019}, the asymptotic behavior of the principal eigenvalue of the above eigenvalue problem as $D\to0$ or $D\to\infty$ was studied. In this paper, when $N\geq2$, under proper conditions on the advection function $m$, we establish the asymptotic behavior of the principal eigenvalue as $α\to\infty$, and when $N=1$, we obtain a complete characterization for such asymptotic behavior provided $m'$ 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}.