A direct approach to Plateau's problem in any codimension

Abstract

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).

Solutions to the Reifenberg Plateau problem with cohomological spanning conditions

“…In 2014 [HP14], the authors proposed usingČech cohomology to extend [HP13] to arbitrary dimension and codimension and elliptic integrands, although no details were given of the latter. In 2015, De Philippis, De Rosa, and Ghiraldin [DPRG15] built upon [HP13] and [DLGM15] and used a relaxed deformation requirement to solve the problem in arbitrary codimension.…”

A direct approach to the anisotropic Plateau problem

“…In [DPDRG15], a similar theorem for the area functional was proved in any codimension. The most general case of any codimension and anisotropic energies will be addressed in a further paper by the same authors, see [DPDRG17], using however different and more sophisticated PDE techniques [DPDRG16].…”

“…Consider the map P ∈ D(0, r) defined in [DPDRG15,Equation 3.14] which collapses R r(1− √ ε),εr onto the tangent plane T K and satisfies P − Id ∞ + Lip (P − Id) ≤ C √ ε. Exploiting the fact that P(H) is a deformation class and by almost minimality of K j , we find that…”

“…An important existence, regularity and almost uniqueness result in arbitrary dimension and codimension was achieved by Almgren in [Alm68], using refined techniques from geometric measure theory. In more recent times, the Plateau problem for the area integrand has been investigated in order to give an existence theory which could comprehend several notions of competitors all together [Feu09,HP13,Dav14,DGM14,DPDRG15]. In particular a very elegant notion of boundary for general closed sets has been introduced by Harrison and Pugh in [HP13], which proves the existence and regularity for minimizers of the area functional in codimension 1.…”

“…In this paper, we adopt the same strategy as in [DGM14,DPDRG15], namely we prove a general compactness theorem for minimizing sequences in general classes of rectifiable sets. More precisely, we consider the measures naturally associated to any such sequence and we show that, if a sufficiently large class of deformations are admitted, any weak limit is induced by a rectifiable set, thus providing compactness and semicontinuity under very little assumptions.…”