Asymptotics of the principal eigenvalue of a linear elliptic operator with large advection
Rui Peng, Guanghui Zhang
TL;DR
This paper analyzes the asymptotic behavior of the principal eigenvalue $λ₁(α)$ for a linear elliptic operator with large advection $-DΔφ-2α∇m·∇φ+Vφ=λφ$ under Robin or Dirichlet boundaries on a bounded smooth domain. The authors develop an energy-and-variational framework, using the transformation $w=e^{αm}φ$ and trace-type inequalities to control boundary contributions, and they characterize the limit of $λ₁(α)$ as $α→∞$ in terms of where the advection function $m$ attains local maxima (interior, boundary with $β=0$, boundary with $β<0$). The main results establish precise limits: in higher dimensions ($N≥2$) the limit is determined by the minimum of $V$ over certain sets $Σ_1, Σ_2$, or is infinite when those sets are empty; for the Dirichlet case the limit reduces to $\min_{x∈Σ_1}V(x)$ when that set is nonempty. In one dimension, a complete characterization is obtained under degeneracy of $m$, with the limit expressed as the minimum among a finite family of subproblem eigenvalues, together with explicit dependence on boundary parameters, illustrating how boundary conditions and advection interact to shape the spectrum. These results extend and complement prior work on the large-advection regime and illuminate the critical influence of boundary terms on the principal eigenvalue’s asymptotics.
Abstract
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: $$ \frac{\partial\varphi}{\partial n}+β(x)\varphi=0 \ \ \hbox{ on }\partialΩ, $$ 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}.