Elliptic Equations: An Introductory Course

by Michel Chipot

Overview

The aim of this book is to introduce the reader to different topics of the theory of elliptic partial differential equations by avoiding technicalities and complicated refinements. Apart from the basic theory of equations in divergence form, it includes subjects as singular perturbations, homogenization, computations, asymptotic behavior of problems in cylinders, elliptic systems, nonlinear problems, regularity theory, Navier-Stokes systems, p-Laplace type operators, large solutions, and mountain pass techniques. Just a minimum on Sobolev spaces has been introduced and work on integration on the boundary has been carefully avoided to keep the reader attention focused on the beauty and variety of these issues.

The chapters are relatively independent of each other and can be read or taught separately. Numerous results presented here are original, and have not been published elsewhere. The book will be of interest to graduate students and researchers specializing in partial differential equations.

Product Details

ISBN-13:	9783031541216
Publisher:	Springer International Publishing
Publication date:	07/15/2024
Series:	Birkh�user Advanced Texts Basler Lehrb�cher
Edition description:	Second Edition 2024
Pages:	397
Product dimensions:	6.10(w) x 9.25(h) x (d)

About the Author

Michel Chipot is Professor at the Institute of Mathematics, University of Zürich, Zürich, Switzerland.

Table of Contents

Part I Basic Techniques.- Hilbert Space Techniques.- A Survey of Essential Analysis.- Weak Formulation of Elliptic Problems.- Elliptic Problems in Divergence Form.- Singular Perturbation Problems.- Problems in Large Cylinders.- Periodic Problems.- Homogenization.- Eigenvalues.- Numerical Computations.- Part II More Advanced Theory.- Nonlinear Problems.- L∞-estimates.- Linear Elliptic Systems.- The Stationary Navier—Stokes System.- Some More Spaces.- Regularity Theory.- p-Laplace Type Equations.- The Strong Maximum Principle.- Problems in the Whole Space.- Large Solutions.- Mountain Pass Techniques.

Editorial Reviews

From the reviews:

“The present book is devoted to recent advanced results and methods in the theory of linear and nonlinear elliptic equations and systems. … It is written with great care and is accessible to a large audience including graduate and postgraduate students and researchers in the field of partial differential equations. … In conclusion, the reviewer may recommend the book as a very good reference for those seeking, new, modern, and powerful techniques in the modern approach of nonlinear elliptic partial differential equations.” (Vicenţiu D. Rădulescu, Zentralblatt MATH, Vol. 1171, 2009)

“The book introduces the reader to a broad spectrum of topics in the theory of elliptic partial differential equations in a simple and systematic way. It provides a comprehensive introductory course to the theory, each chapter being supplemented with interesting exercises for the reader. … The way of presentation of the material … keep the reader’s attention on the beauty and variety of the issues. … a very valuable position in the field of elliptic partial differential equations.” (Irena Pawłow, Control and Cybernetics, Vol. 39 (3), 2010)

From the Publisher