Abstract

In this paper, we study the differential inclusion associated with the minimal surface system for two-dimensional graphs in ${\mathbb{R}}^{2+n}.$ We prove regularity of $W^{1,2}$ solutions and a compactness result for approximate solutions of this differential inclusion in $W^{1,p}.$ Moreover, we make a perturbation argument to infer that for every R > 0, there exists $\alpha (R)>0$ such that R-Lipschitz stationary points for functionals α-close in the C² norm to the area functional are always regular. We also use a counterexample of B. Kirchhem (2003) to show the existence of irregular critical points to inner variations of the area functional.

中文翻译：

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最小图形和微分包含

摘要

在本文中，我们研究了与二维图的最小表面系统相关的微分包含 ${\mathbb{电阻}}^{2+n}.$ 我们证明的规律性 ${宽}^{1,2}$ 解和这​​种微分包含的近似解的紧致性结果 ${宽}^{1,磷}.$此外，我们提出了一个扰动论证来推断，对于每一个R  > 0，存在$\alpha (\mathrm{电阻})>0$使得泛函α 的R -Lipschitz 驻点在C ²范数中接近区域泛函总是规则的。我们还使用 B. Kirchhem (2003) 的反例来展示区域函数内部变化的不规则临界点的存在。