  Let $M$ be a weighted manifold with boundary $\partial M$, i.e., a Riemannian
manifold where a density function is used to weight the Riemannian Hausdorff
measures. In this paper we compute the first and the second variational
formulas of the interior weighted area for deformations by hypersurfaces with
boundary in $\partial M$. As a consequence, we obtain variational
characterizations of critical points and second order minima of the weighted
area with or without a volume constraint. Moreover, in the compact case, we
obtain topological estimates and rigidity properties for free boundary stable
and area-minimizing hypersurfaces under certain curvature and boundary
assumptions on $M$. Our results and proofs extend previous ones for Riemannian
manifolds (constant densities) and for hypersurfaces with empty boundary in
weighted manifolds.
