General Methods of Elliptic Minimization

TL;DR

This paper introduces new general methods in the calculus of variations to solve the anisotropic Plateau problem across various dimensions and codimensions, providing a direct proof of existence and rectifiability of minimizers.

Contribution

It presents a novel direct proof of Almgren's existence theorem, extends the setting to Lipschitz neighborhood retracts, and introduces an axiomatic spanning criterion for anisotropic minimal surfaces.

Findings

Established rectifiability of minimizers without quasiminimality assumptions.

Provided a new direct proof of Almgren's 1968 existence result.

Extended the ambient space to include manifolds with boundary and singularities.

Abstract

We provide new general methods in the calculus of variations for the anisotropic Plateau problem in arbitrary dimension and codimension. A new direct proof of Almgren's 1968 existence result is presented; namely, we produce from a class of competing "surfaces," which span a given bounding set in some ambient space, one with minimal anisotropically weighted area. In particular, rectifiability of a candidate minimizer is proved without the assumption of quasiminimality. Our ambient spaces are a class of Lipschitz neighborhood retracts which includes manifolds with boundary and manifolds with certain singularities. Our competing surfaces are rectifiable sets which satisfy any combination of general homological, cohomological or homotopical spanning conditions. An axiomatic spanning criterion is also provided. Our boundaries are permitted to be arbitrary closed subsets of the ambient space, providing a good setting for surfaces with sliding boundaries.