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Lectures on Differential Equations and Differential Geometry

Louis Nirenberg
Publisher: 
Higher Education Press
Publication Date: 
2019
Number of Pages: 
174
Format: 
Hardcover
Series: 
Classical Topics in Mathematics 7
Price: 
59.00
ISBN: 
9787040503029
Category: 
Monograph
[Reviewed by
Bill Satzer
, on
04/6/2019
]

This volume includes two distinct sets of lecture notes prepared by Louis Nirenberg and published as part of the Classical Topics in Mathematics series. Although the book was published recently, the lecture notes likely date from many years ago. The current volume provides no information about the date they were actually written (and the internet provided no additional information), but it is possible that these may have appeared in the NYU Courant lecture note series as early as the 1970s or even before.

Whatever their origin, this publication brings back into circulation some elegant work by Louis Nirenberg. Both parts of the book are notable for the quality of their exposition, and both continue to have value for specialists in partial differential equations and differential geometry and well-prepared graduate students. Nirenberg focuses here on notable results in areas of particular interest to him. This is not a textbook, and there are no exercises.

Readers interested in broader treatments of PDEs would best look to standard textbooks in those areas, such as the book reviewed here and the standard texts cited in that review. The current book assumes a level of experience with PDEs that makes it unsuitable for those without a fairly strong background.

The first part of the book concentrates on existence and uniqueness problems for PDEs, especially elliptic equations and their associated boundary value problems. Nirenberg’s strategy is first to study the Dirichlet boundary value problem for linear elliptic equations. He wants very sharp estimates for linear equations to provide tools to attack the more challenging second order nonlinear equations, and these he gets in part via Schauder estimates. The Laplace and Poisson equations are given special treatment.

The second set of lectures address differential geometry “in the large”. This relates to work that Nirenberg did in the 1950s, and it includes his famous work on the Minkowski problem: to determine a closed convex surface with a given Gaussian curvature assigned as a continuous function of the interior normal to the surface. Nirenberg’s approach reduces the problem to questions about nonlinear elliptic PDEs. Much of the material in these lectures – especially the work on surfaces - is not available in standard texts on differential geometry.

These lectures also include results about the rigidity of convex surfaces. A main result here is that if a closed convex surface in three-space is deformed continuously and isometrically into another closed convex surface, then the deformation is essentially equivalent to a rigid body motion.

The writing throughout the book is smooth and tightly focused with little in the way of elaboration and no digressions. Nirenberg always seems to take the most direct possible path to his results. He doesn’t make many concessions to the reader. The only references provided are the few that appear in footnotes. There is no index.


Bill Satzer (bsatzer@gmail.com) was a senior intellectual property scientist at 3M Company. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

 

The table of contents is not available.

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