World Scientific Monograph Series in Mathematics - Vol. 1
ALMGREN'S BIG REGULARITY PAPER
Q-Valued Functions Minimizing Dirichlet's Integral and the Regularity of Area-Minimizing Rectifiable Currents up to Codimension 2
edited by Vladimir Scheffer & Jean E Taylor (Rutgers University)
Fred Almgren exploited the excess method for proving regularity theorems in the calculus of variations. His techniques yielded Hölder 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.
Contents:
- Basic Properties of Q and Q Valued Functions
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Properties of Dir-Minimizing Q Valued Functions and Tangent Cone Stratification of Mass Minimizing Rectifiable Currents
- Approximation in Mass of Nearly Flat Rectifiable Currents which are Mass Minimizing in Manifolds by Graphs of Lipschitz Q Valued Functions Which Can Be Weakly Nearly Dir Minimizing
- Approximation in Mass of a Nearly Flat Rectifiable Current Which Is Mass Minimizing in a Manifold by the Image of a Lipschitz Q(Rm+n) Valued Function Defined on a Center Manifold
- Bounds on the Frequency Functions and the Main Interior Regularity Theorem
Readership: Students and researchers dealing with the calculus of
variations.
"The book closes with a number of appendices which also are of independent interest, and it starts with a beautiful Introduction (16 pages) which contains a 'Summary of the principal themes' by chapters ... This work is a monument."
"Now, thanks to the efforts of editors Jean Taylor and Vladimir Scheffer, Almgren's three-volume, 1700-page typed preprint has been published as a single, attractively typset volume of less than 1000 pages ... Perhaps advances in knowledge will eventually make possible a much shorter and more transparent proof of Almgren's theorem. But I suspect that if such a proof is discovered, it will still use the basic approach and many of the tools pioneered by Almgren in this monumental work."
| 972pp |
Pub. date: Jul 2000 |
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