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

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*} -\Delta\varphi(x)-2s\mathbf{v}\cdot\nabla\varphi(x)+c(x)\varphi(x)=\lambda(s)\varphi(x),\quad x\in \Omega, \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.