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We solve two variants of the Reifenberg problem for all coefficient groups. We carry out the direct method of the calculus of variation and search a solution as a weak limit of a minimizing sequence. This strategy has been introduced by De Lellis, De Philippis, De Rosa, Ghiraldin and Maggi and allowed them to solve the Reifenberg problem. We use an analogous strategy proved in [C. Labourie, Weak limits of quasiminimizing sequences, preprint 2020,  https://arxiv.org/abs/2002.08876 ] which allows to take into account the free boundary. Moreover, we show that the Reifenberg class is closed under weak convergence without restriction on the coefficient group.

Solutions of the (free boundary) Reifenberg Plateau problem