Equivalence of the ellipticity conditions for geometric variational problems

TL;DR

This paper demonstrates that the atomic condition, recently introduced for anisotropic geometric measure theory, implies the strict Almgren ellipticity condition, clarifying the relationship between these key ellipticity criteria.

Contribution

It proves that the atomic condition implies the strict Almgren geometric ellipticity condition, resolving an open question in anisotropic geometric measure theory.

Findings

Atomic condition implies strict Almgren ellipticity

Addresses an open question in the field

Strengthens understanding of ellipticity conditions in variational problems

Abstract

We exploit the so called atomic condition, recently defined by De Philippis, De Rosa, and Ghiraldin in [Comm. Pure Appl. Math.] and proved to be necessary and sufficient for the validity of the anisotropic counterpart of the Allard rectifiability theorem. In particular, we address an open question of this seminal work, showing that the atomic condition implies the strict Almgren geometric ellipticity condition.