  We introduce and study co-dimension one area-minimizing locally rectifiable
currents $T$ with $C^{1,\alpha}$ tangentially immersed boundary: $\partial T$
is locally a finite sum of orientable co-dimension two submanifolds which only
intersect tangentially with equal orientation. We show that any such $T$ is
supported in a smooth hypersurface near any point on the support of $\partial
T$ where $T$ has tangent cone which is a hyperplane with constant orientation
but non-constant multiplicity. We also introduce and study co-dimensional one
area-minimizing locally rectifiable currents $T$ with boundary having
co-oriented mean curvature: $\partial T$ has generalized mean curvature
$H_{\partial T} = h \nu_{T}$ with $h$ a real-valued function and $\nu_{T}$ the
generalized outward pointing unit normal of $\partial T$ with respect to $T.$
