Geometry of the Atomic Condition

Abstract

The class AC of integrands satisfying the atomic condition was introduced by De Philippis, De Rosa, and Ghiraldin in 2018. These integrands give rise to Almgren elliptic geometric functionals as proven by the second author and De Rosa in 2020. So far, it is not known how to verify the atomic condition for any particular integrand (apart from the area integrand and its class 2 neighbourhood) or how to construct, event artificial, members of AC. We reinterpret the atomic condition in terms of convex geometry shedding light on the structure of this class. We also propose quantitative versions of AC, different from the SAC and USAC defined by De Rosa and Tione, which we call the exposed condition EC and the quadratic exposed condition QEC. As is the case with USAC, the QEC is stable under class 2 perturbations and supports a Caccioppoli-type inequality; hence, is suitable for proving regularity of critical points. Moreover, we provide some conditions sufficient for EC in case the integrand is associated to a norm on the exterior power of $\mathbf{R}^{n}$. Finally, for all $k \ge 2$ and $n-k \ge 3$ we construct strictly polyconvex integrands -- associated to inner-product norms on $\bigwedge_{k} \mathbf{R}^{n}$ -- which fail the atomic condition, showing that strict polyconvexity is necessary but far from sufficient for AC.