Existence of solutions to a general geometric elliptic variational problem

Abstract

We consider the problem of minimising an inhomogeneous anisotropic elliptic functional in a class of closed $m$ dimensional subsets of $\mathbf{R}^n$ 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 $(\mathscr{H}^m,m)$~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 $1 \le m < n$. An important tool for the proof is a novel smooth deformation theorem. The skeleton of the proof and the main ideas follow Almgren's 1968 paper. In the end we show that classes of sets spanning some closed set $B$ in homological and cohomological sense satisfy our axioms.