In this paper we prove sharp regularity for a differential inclusion into a set $$K\subset {{\mathbb {R}}}^{2\times 2}$$ K ⊂ R 2 × 2 that arises in connection with the Aviles–Giga functional. The set K is not elliptic, and in that sense our main result goes beyond Šverák’s regularity theorem on elliptic differential inclusions. It can also be reformulated as a sharp regularity result for a critical nonlinear Beltrami equation. In terms of the Aviles–Giga energy, our main result implies that zero energy states coincide (modulo a canonical transformation) with solutions of the differential inclusion into K . This opens new perspectives towards understanding energy concentration properties for Aviles–Giga: quantitative estimates for the stability of zero energy states can now be approached from the point of view of stability estimates for differential inclusions. All these reformulations of our results are strong improvements upon a recent work by the last two authors, Lorent and Peng, where the link between the differential inclusion into K and the Aviles–Giga functional was first observed and used. Our proof relies moreover on new observations concerning the algebraic structure of entropies.

中文翻译：

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与 Aviles-Giga 猜想相关的非椭圆微分包含的刚度

在本文中，我们证明了与 Aviles 相关的微分包含到集合 $$K\subset {{\mathbb {R}}}^{2\times 2}$$ K ⊂ R 2 × 2 中的锐利规律性– 千兆功能。集合 K 不是椭圆的，从这个意义上说，我们的主要结果超出了 Šverák 关于椭圆微分包含的正则性定理。它也可以重新表述为临界非线性贝尔特拉米方程的锐利规律性结果。就 Aviles-Giga 能量而言，我们的主要结果意味着零能量状态与 K 的微分包含的解重合（以正则变换为模）。这为理解 Aviles-Giga 的能量集中特性开辟了新的视角：现在可以从微分夹杂物的稳定性估计的角度对零能量状态的稳定性进行定量估计。我们结果的所有这些重新表述都是对最后两位作者 Lorent 和 Peng 的最近工作的有力改进，他们首先观察并使用了 K 中的微分包含与 Aviles-Giga 泛函之间的联系。我们的证明还依赖于关于熵的代数结构的新观察。