A Counterexample for the Principal Eigenvalue of An Elliptic Operator with Large Advection

Abstract

There are numerous studies focusing on the convergence of the principal eigenvalue $\lambda (s)$ as $s\to +\infty$ corresponding to the elliptic eigenvalue problem \begin{align*} -Δφ(x)-2s\mathbf{v}\cdot\nablaφ(x)+c(x)φ(x)=λ(s)φ(x),\quad x\in Ω, \end{align*} where $\Omega$ is a bounded domain and the advection term $\mathbf{v}$ under some certain restrictions. In this paper, we construct an infinitely oscillating gradient advection term $\mathbf{v}=\nabla m(x)\in C^1(\Omega )$ such that the principal eigenvalue $\lambda (s)$ does not converge as $s\to +\infty$. As far as we know, this is the first result that guarantee the non-convergence of the principal eigenvalue.