Given an open set Ω ⊂ R m and n > 1, we introduce the new spaces GBnV (Ω) of Generalized functions of bounded higher variation and GSBnV (Ω) of Generalized special functions of bounded higher variation that generalize, respectively, the space BnV introduced by Jerrard and Soner in [43] and the corresponding SBnV space studied by De Lellis in [24]. In this class of spaces, which allow as in [43] the description of singularities of codimension n, the distributional jacobian Ju need not have finite mass: roughly speaking, finiteness of mass is not required for the (m − n)-dimensional part of Ju, but only finiteness of size. In the space GSBnV we are able to provide compactness of sublevel sets and lower semicontinuity of Mumford-Shah type functionals, in the same spirit of the codimension 1 theory [5,6].

We provide a compactness principle which is applicable to different formulations of Plateau's problem in codimension one and which is exclusively based on the theory of Radon measures and elementary comparison arguments. Exploiting some additional techniques in geometric measure theory, we can use this principle to give a different proof of a theorem by Harrison and Pugh and to answer a question raised by Guy David.

Abstract. This paper aims to propose a direct approach to solve the Plateau's problem in codimension higher than one. The problem is formulated as the minimization of the Hausdorff measure among a family of d-rectifiable closed subsets of R n : following the previous work [DGM14] the existence result is obtained by a compactness principle valid under fairly general assumptions on the class of competitors. Such class is then specified to give meaning to boundary conditions. We also show that the obtained minimizers are regular up to a set of dimension less than (d − 1).

We extend Allard's celebrated rectifiability theorem to the setting of varifolds with locally bounded first variation with respect to an anisotropic integrand. In particular, we identify a sufficient and necessary condition on the integrand to obtain the rectifiability of every d -dimensional varifold with locally bounded first variation and positive d -dimensional density. In codimension 1, this condition is shown to be equivalent to the strict convexity of the integrand with respect to the tangent plane.