AI 中文总结
研究满足原子条件的被积函数类AC，用凸几何重新解释原子条件并提出定量版本EC和QEC，给出EC充分条件，还构造不满足原子条件的严格多凸被积函数，揭示严格多凸性对AC必要但不充分。
AI 中文摘要
2018年，德菲利普斯、德罗萨和吉拉尔丁引入了满足原子条件的被积函数类AC。2020年，第二作者和德罗萨证明这些被积函数会产生阿尔姆格伦椭圆几何泛函。目前尚不清楚如何验证任何特定被积函数的原子条件（除了面积被积函数及其2类邻域），也不知道如何构造AC的成员。我们用凸几何重新解释原子条件，揭示了这类的结构。我们还提出了AC的定量版本，不同于德罗斯和蒂奥内定义的SAC和USAC，我们称之为暴露条件EC和二次暴露条件QEC。与USAC一样，QEC在2类扰动下是稳定的，并支持一个卡乔波利型不等式，因此适用于证明临界点的正则性。此外，我们给出了被积函数与\(\mathbf{R}^{n}\)的外幂上的范数相关时EC的一些充分条件。最后，对于所有\(k \ge 2\)和\(n - k \ge 3\)，我们构造了严格多凸被积函数（与\(\bigwedge_{k} \mathbf{R}^{n}\)上的内积范数相关），它们不满足原子条件，表明严格多凸性对于AC是必要的，但远远不够。
英文摘要
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.
