Variational problems of least area type with constraints

Summary We establish a new regularity result for boundaries of sets minimizing the area functional perturbed by a curvature term, and subject to a mass constraint of the type , with a fixed, strictly positive integrable functiong(x). The existence of Lagrange parameters is also considered. A preliminary version of the paper was presented at the meeting on “Partial Differential Equations”, Ferrara (Italy), October 13–16, 1986. Work partially supported by the Italian M.P.I.