Geometric Measure Theory: A Beginner's Guide

by Frank Morgan

Overview

Geometric measure theory is the mathematical framework for the study of crystal growth, the geometry of clusters of soap bubbles, and similar structures involving minimization of surface energy. Beautiful mathematics comes into play when a group of bubbles forms a cluster and physically solves the complicated mathematical problem of minimizing surface energy.

Product Details

ISBN-13:	9780128045275
Publisher:	Elsevier Science
Publication date:	05/02/2016
Sold by:	Barnes & Noble
Format:	eBook
Pages:	272
File size:	11 MB
Note:	This product may take a few minutes to download.

About the Author

Frank Morgan is the Dennis Meenan '54 Third Century Professor of Mathematics at Williams College. He obtained his B.S. from MIT and his M.S. and Ph.D. from Princeton University. His research interest lies in minimal surfaces, studying the behavior and structure of minimizers in various settings. He has also written Riemannian Geometry: A Beginner's Guide, Calculus Lite, and most recently The Math Chat Book, based on his television program and column on the Mathematical Association of America Web site.

Table of Contents

Preface. Geometric Measure Theory. Measures. Lipschitz Functions and Rectifiable Sets. Normal and Rectifiable Currents. The Compactness Theorem and the Existence of Area-Minimizing Surfaces. Examples of Area-Minimizing Surfaces. The Approximation Theorem. Survey of Regularity Results. Monotonicity and Oriented Tangent Cones. The Regularity of Area-Minimizing Hypersurfaces. Flat Chains Modulo v, Varifolds, and (M, E, )-Minimal Sets. Miscellaneous Useful Results. Soap Bubble Clusters. Proof of Double Bubble Conjecture. The Hexagonal Honeycomb and Kelvin Conjectures. Immiscible Fluids and Crystals. Isoperimetric Theorems in General Codimension. Solutions to Exercises. Bibliography. Index of Symbols. Name Index. Subject Index.

Editorial Reviews

This graduate textbook introduces the basic ideas, terminology, and results of geometric measure theory for students who have taken a course in real analysis. The third edition adds four chapters that present proofs of the Double Bubble and Hexagonal Honeycomb conjectures and recent results in immiscible fluids and isoperimetric inequalities. Annotation c. Book News, Inc., Portland, OR (booknews.com)

Booknews

"Morgan’s book offers the best access I know into this difficult subject. It won’t make the reader an expert, but it does open the door to more detailed treatments. An excellent bibliography is provided." —MAA Reviews, Geometric Measure Theory, Fifth Edition

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