Regularity Effects on the Small-Diffusion Asymptotics of the Principal Eigenvalue: Non-convergence and Convergence

We study the asymptotic behavior as $d\to0^+$ of the principal eigenvalue $λ(d)$ of the one-dimensional elliptic problem $\begin{cases} -dϕ''(x)-2αm'(x)ϕ'(x)+c(x)ϕ(x) =λ(d)ϕ(x), & x\in(a,b),\\ ϕ'(a)=ϕ'(b)=0, \end{cases}$ where $α>0$ is fixed, $m\in C^{1,1}([a,b])$, and $c\in C([a,b])$. We assume that $m'=0$ only at $x_0\in(a,b)$, where $m$ attains a strict minimum. Within this class, we construct a counterexample $m\in C^{1,1}$ for which $λ(d)$ has two distinct subsequential limits as $d\to0^+$, leading to the non-convergence of $λ(d)$. We then consider another class of $C^{1,1}$ advection functions for which $m'(x)/(x-x_0)$ has positive one-sided limits at $x_0$. For this class, we prove that $λ(d)$ converges as $d\to0^+$. Thus, $C^{1,1}$ regularity alone does not ensure convergence of the principal eigenvalue, even when $m$ has exactly one strict interior critical minimal point.