Let $m, s \in ([0, 1]^d \rightarrow \mathbb{R}_{\geq 0}$). Define a weighted Laplacian $\Delta_{m, s}f$ evaluated at $x \in [0, 1]^d$ to be:
$m(x) \cdot \text{div} ( s(x) \nabla f(x))$.
What conditions do we need on $m$ and $s$ to be able to say: If $-\Delta_{m, s} f = \lambda f$ for the smallest non-zero eigenvalue $\lambda$, and $f$ is zero on an open set, then $f$ is identically zero?
I'm particularly interested in the case when $m, s$ are continuous 1-Lipschitz functions bounded below, but any condition on $m$ and $s$ (like analyticness or smoothness) would be helpful.
(Here, when $m(x) = s(x) = 1$ and $[0,1]^d$ is replaced with $\mathbb{R}^d$, $\Delta_{m, s}$ is the usual Laplacian, https://math.stackexchange.com/questions/672383/if-a-laplacian-eigenfunction-is-zero-in-an-open-set-is-it-identically-zero holds, and we don't need to use the fact that $\lambda$ is smallest.)
