Consider the regular Sturm-Liouville eigenvalue equation $$ \frac{d}{dx}(p_t(x)f^\prime(x))=\lambda_t f(x) $$ for $p_t\in\mathcal{C}^\infty([0,1])$ with Dirichlet boundary conditions on $[0,1]$. Here $t\geq 0$ is a parameter and let's say $t\mapsto p_t(x)$ is smooth for all $x\in[0,1]$ and $p_0=1$. Denote the smallest eigenvalue $\lambda_t$ by $\lambda_t^0$. So $\lambda_0^0=\pi^2$.

Can anything be said about the mapping $t\mapsto \lambda_t^0$? E.g. is it monotone in $t$ (likely to depend on $p_t$), what's its scaling behavior, or is it smooth?