Rigidity of a Non-elliptic Differential Inclusion Related to the Aviles–Giga Conjecture
Rigidity of a Non-elliptic Differential Inclusion Related to the Aviles–Giga Conjecture
复制标题
与阿维莱斯-吉加猜想相关的非椭圆微分包含的刚性
DOI：
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发表时间：
2019
影响因子：
2.5
通讯作者：
G. Peng
中科院分区：
文献类型：
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作者：
X. Lamy;A. Lorent;G. Peng
在本文中，我们证明了差分包含到集合 K⊂R2×2documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt}egin{document}$$Ksubset 中的尖锐规律性{{mathbb {R}}}^{2 imes 2}$$end{document} 与 Aviles-Giga 泛函相关。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定理。它也可以重新表述为临界非线性贝尔特拉米方程的尖锐正则性结果。就 Aviles-Giga 能量而言，我们的主要结果意味着零能态与 K 的微分包含体的解一致（模正则变换）。这为理解 Aviles-Giga 的能量集中特性开辟了新的视角：现在可以从微分包含体的稳定性估计的角度来对零能态的稳定性进行定量估计。我们对结果的所有这些重新表述都是对最后两位作者 Lorent 和 Peng 最近的工作的重大改进，其中首次观察并使用了 K 的微分包含与 Aviles-Giga 泛函之间的联系。此外，我们的证明依赖于关于熵代数结构的新观察。
In this paper we prove sharp regularity for a differential inclusion into a set K⊂R2×2documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$Ksubset {{mathbb {R}}}^{2 imes 2}$$end{document} 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.