Existence of solutions to a general geometric elliptic variational problem

We consider the problem of minimising an inhomogeneous anisotropic elliptic functional in a class of closed m dimensional subsets of R n which is stable under taking smooth deformations homotopic to the identity and under local Hausdorff limits. We prove that the minimiser exists inside the class and is an ( H m , m) rectifiable set in the sense of Federer. The class of competitors encodes a notion of spanning a boundary. We admit unrectifiable and non-compact competitors and boundaries, and we make no restrictions on the dimension m and the co-dimension n − m other than 1 ≤ m < n. An important tool for the proof is a novel smooth deformation theorem. The skeleton of the proof and the main ideas follow Almgren’s (Ann Math (2) 87:321–391, 1968) paper. In the end we show that classes of sets spanning some closed set B in homological and cohomological sense satisfy our axioms.