Geometry of the Atomic Condition
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. Explicit examples in intermediate dimensions include small class 2 perturbations of the area integrand and the recent anisotropies constructed by De Rosa, Firester, and Tsiamis, among them the $\ell^q$ norms in Plücker coordinates for $q$ close to $2$. We reinterpret the atomic condition in terms of convex geometry. We also propose variants of AC, related to the SAC and USAC defined by De Rosa and Tione, which we call the exposed condition EC, the strong exposed condition SEC, and the quadratic exposed condition QEC. For integrands of class 2, QEC is equivalent to validity of a Caccioppoli-type inequality and is stable under class 2 perturbations; every such QEC integrand admits Lipschitzian normals. Combining these results with a result of De Rosa, Firester, and Tsiamis, we prove that a Lipschitz graph whose anisotropic mean curvature for a QEC integrand of class 2 is $p$-integrable, with $k < p < \infty$, is of class $\mathscr{C}^{1,α}$ on an open set of full measure. 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 not sufficient for AC.