  For a given domain $\Omega \subset \Bbb{R}^n$, we consider the variational
problem of minimizing the $L^1$-norm of the gradient on $\Omega$ of a function
$u$ with prescribed continuous boundary values and satisfying a continuous
lower obstacle condition $u\ge \Psi$ inside $\Omega$. Under the assumption of
strictly positive mean curvature of the boundary $\partial\Omega$, we show
existence of a continuous solution, with H\"older exponent half of that of data
and obstacle.
  This generalizes previous results obtained for the unconstrained and
double-obstacle problems. The main new feature in the present analysis is the
need to extend various maximum principles from the case of two area-minimizing
sets to the case of one sub- and one superminimizing set. This we accomplish
subject to a weak regularity assumption on one of the sets, sufficient to carry
out the analysis. Interesting open questions include the uniqueness of
solutions and a complete analysis of the regularity properties of area
superminimizing sets. We provide some preliminary results in the latter
direction, namely a new monotonicity principle for superminimizing sets, and
the existence of ``foamy'' superminimizers in two dimensions.
