| Abstract: |
| In this talk, we would like to report our recent work on the the convergence of the principal eigenvalue of the elliptic operator
\begin{equation*}
-d\Delta \varphi(x)-2 s v(x)\cdot \nabla \varphi(x)+c(x)\varphi(x)=\lambda(d,s)\varphi(x)
\end{equation*}
Firstly, we constructed an infinitely oscillating gradient advection term such that the principal eigenvalue does not converge as $s\to+\infty$. Secondly, we considered the principal eigenvalue of the elliptic operator with a constant advection term and the boundary conditions $\varphi'(0)=s(1+b_0)\varphi(0)$ and $\varphi'(1)=s(1-b_1)\varphi(1)$ in one dimension. Then we determined the limits of these principal eigenvalues as $s$ tends to infinity for all parameters $b_0,b_1\in(-\infty,+\infty)$. Recently, we have further investigated the asymptotic eigenvalue problem on manifolds and obtained some results. |
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