  We prove the existence of a continuous $BV$ minimizer with $C^{0}$ boundary
value for the $p$-area (pseudohermitian or horizontal area) in a parabolically
convex bounded domain. We extend the domain of the area functional from $BV$
functions to vector-valued measures. Our main purpose is to study the first and
second variations of such a generalized area functional including the
contribution of the singular part. By giving examples in Riemannian and
pseudohermitian geometries, we illustrate several known results in a unified
way. We show the contribution of the singular curve in the first and second
variations of the $p$-area for a surface in an arbitrary pseudohermitian
$3$-manifold.
