Almgren's big regularity paper : Q-valued functions minimizing Dirichlet's integral and the regularity of area-minimizing rectifiable currents up to codimension 2
Almgren's big regularity paper : Q-valued functions minimizing Dirichlet's integral and the regularity of area-minimizing rectifiable currents up to codimension 2
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DOI：
10.1142/4253
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发表时间：
2000-06
期刊：
影响因子：
--
通讯作者：
Vladimir Scheffer;Jean E. Taylor
中科院分区：
文献类型：
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作者：
Vladimir Scheffer;Jean E. Taylor
弗雷德·阿尔姆格伦利用过量法证明了变分学中的正则定理。他的技术得到了除了一个小的闭奇异集以外的Holder连续可微性。在六七十年代，Almgren改进并推广了他的方法。1974年至1984年间，他写了1700页的证明，这是他对自己突破性思想最雄心勃勃的发展。最初，这本专著只作为一个三卷的有限流通的工作。全文原封不动地复制在这里。这本书给出了一个面积最小的可整流电流达到豪斯多夫余维数2的内部正则性的完整证明。论证采用了q值函数理论，并对其进行了详细的阐述。例如，这项工作显示了如何从挤压变形和挤压变形的第一变差估计产生一个单调性定理，该定理用于最小化广义狄利克雷积分的q值函数的归一化振荡频率。本书的主要特点包括一个类似于柯兹布劳恩定理的扩展定理，以及用李普希茨q值函数的图和图像逼近近平坦的质量最小化可整流的定理。
Fred Almgren exploited the excess method for proving regularity theorems in the calculus of variations. His techniques yielded Holder continuous differentiability except for a small closed singular set. In the sixties and seventies Almgren refined and generalized his methods. Between 1974 and 1984 he wrote a 1,700-page proof that was his most ambitious development of his ground-breaking ideas. Originally, this monograph was available only as a three-volume work of limited circulation. The entire text is faithfully reproduced here.This book gives a complete proof of the interior regularity of an area-minimizing rectifiable current up to Hausdorff codimension 2. The argument uses the theory of Q-valued functions, which is developed in detail. For example, this work shows how first variation estimates from squash and squeeze deformations yield a monotonicity theorem for the normalized frequency of oscillation of a Q-valued function that minimizes a generalized Dirichlet integral. The principal features of the book include an extension theorem analogous to Kirszbraun's theorem and theorems on the approximation in mass of nearly flat mass-minimizing rectifiable currents by graphs and images of Lipschitz Q-valued functions.