![]() ![]() Nirenberg’s approach reduces the problem to questions about nonlinear elliptic PDEs. 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. The second set of lectures address differential geometry “in the large”. The Laplace and Poisson equations are given special treatment. 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. Nirenberg’s strategy is first to study the Dirichlet boundary value problem for linear elliptic equations. The first part of the book concentrates on existence and uniqueness problems for PDEs, especially elliptic equations and their associated boundary value problems. The current book assumes a level of experience with PDEs that makes it unsuitable for those without a fairly strong background. 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. This is not a textbook, and there are no exercises. Nirenberg focuses here on notable results in areas of particular interest to him. 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. ![]() Whatever their origin, this publication brings back into circulation some elegant work by Louis Nirenberg. 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. Although the book was published recently, the lecture notes likely date from many years ago. In this attractive, inexpensive paperback edition, it belongs in the library of any mathematician or student of mathematics interested in differential geometry.This volume includes two distinct sets of lecture notes prepared by Louis Nirenberg and published as part of the Classical Topics in Mathematics series. The result was to further increase the merit of this stimulating, thought-provoking text - ideal for classroom use, but also perfectly suited for self-study. Struik has enhanced the treatment with copious historical, biographical, and bibliographical references that place the theory in context and encourage the student to consult original sources and discover additional important ideas there.įor this second edition, Professor Struik made some corrections and added an appendix with a sketch of the application of Cartan's method of Pfaffians to curve and surface theory. Written by a noted mathematician and historian of mathematics, this volume presents the fundamental conceptions of the theory of curves and surfaces and applies them to a number of examples. A selection of more difficult problems has been included to challenge the ambitious student. The text features an abundance of problems, most of which are simple enough for class use, and often convey an interesting geometrical fact. It is aimed at advanced undergraduate and graduate students who will find it not only highly readable but replete with illustrations carefully selected to help stimulate the student's visual understanding of geometry. Elementary, yet authoritative and scholarly, this book offers an excellent brief introduction to the classical theory of differential geometry.
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