Solutions To Lipschitz Variational Problems With Cohomological Spanning Conditions

We prove existence and regularity of minimizers for Lipschitz integrands over general surfaces of arbitrary dimension and codimension in \( \R^n \), satisfying a cohomological boundary condition. Our result specializes to a version of Plateau's problem in the case of a constant integrand. We generalize and extend methods of Reifenberg, Besicovitch, and Adams; in particular, we prove an asymptotic monotonicity result (density exists in the absence of true monotonicity,) we generalize a type of minimizing sequence used by Reifenberg (whose limits have nice properties, including lower bounds on lower density and finite Hausdorff measure,) and develop cohomological spanning conditions.