Here we study the regularity or smoothness (in the interior or at the boundary) of an n-dimensional hypersurface in R('n+1) which minimizes area (or more generally an integral of a positive parametric elliptic integrand) and which satisfies certain boundary and obstacle constraints. First, assuming certain smoothness on the boundary and obstacle, we show when, in the context of geometric measure theory, the minimizer is, near the obstacle, a smooth submanifold with boundary. Second, assuming very little smoothness of the obstacle, we study the regularity of a minimizing graph for a corresponding problem in the contex of partial differential equations and variational equalities. The methods developed here are also useful for some questions concerning the solvability of the classical Dirichlet problem for minimal-surface-type systems with given small boundary data. Finally we develop a perturbation theory for immersed hypersurfaces of prescribed mean curvature.
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