Partial Differential Equations III: Nonlinear Equations (Applied Mathematical Sciences, 117) by Michael E. Taylor (PDF)

Description

The third of three volumes on partial differential equations, this is devoted to nonlinear PDE. It treats a number of equations of classical continuum mechanics, including relativistic versions, as well as various equations arising in differential geometry, such as in the study of minimal surfaces, isometric imbedding, conformal deformation, harmonic maps, and prescribed Gauss curvature. In addition, some nonlinear diffusion problems are studied. It also introduces such analytical tools as the theory of L Sobolev spaces, H lder spaces, Hardy spaces, and Morrey spaces, and also a development of Calderon-Zygmund theory and paradifferential operator calculus. The book is aimed at graduate students in mathematics, and at professional mathematicians with an interest in partial differential equations, mathematical physics, differential geometry, harmonic analysis and complex analysis

User’s Reviews

Editorial Reviews: Review From the reviews:“These volumes will be read by several generations of readers eager to learn the modern theory of partial differential equations of mathematical physics and the analysis in which this theory is rooted.”(SIAM Review, June 1998)From the reviews of the second edition:“This substantial three-volume work is an upgraded version of the comprehensive qualitative analysis of partial differential equations presented in the earlier edition. … Graduate students … will find these three volumes to be not just a fine and rigorous treatment of the subject, but also a source of inspiration to apply their knowledge and ability to the solution of other challenging problems in the field of partial differential equations. … an excellent text for all devotees of the charming and thought-provoking byways of higher mathematics.” (Christian Constanda, The Mathematical Association of America, July, 2011) From the Back Cover The third of three volumes on partial differential equations, this is devoted to nonlinear PDE. It treats a number of equations of classical continuum mechanics, including relativistic versions, as well as various equations arising in differential geometry, such as in the study of minimal surfaces, isometric imbedding, conformal deformation, harmonic maps, and prescribed Gauss curvature. In addition, some nonlinear diffusion problems are studied. It also introduces such analytical tools as the theory of L^p Sobolev spaces, Holder spaces, Hardy spaces, and Morrey spaces, and also a development of Calderon-Zygmund theory and paradifferential operator calculus. The book is targeted at graduate students in mathematics and at professional mathematicians with an interest in partial differential equations, mathematical physics, differential geometry, harmonic analysis, and complex analysis. In this second edition, there are seven new sections including Sobolev spaces on rough domains, boundary layer phenomena for the heat equation, an extension of complex interpolation theory, and Navier-Stokes equations with small viscosity. In addition, several other sections have been substantially rewritten, and numerous others polished to reflect insights obtained through the use of these books over time. Michael E. Taylor is a Professor of Mathematics at the University of North Carolina, Chapel Hill, NC. Review of first edition: “These volumes will be read by several generations of readers eager to learn the modern theory of partial differential equations of mathematical physics and the analysis in which this theory is rooted.”(SIAM Review, June 1998) About the Author Michael E. Taylor is a Professor at University of North Carolina in the Department of Mathematics. Read more

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐If you have a nonlinear problem and you are looking for results on local and global existence, uniqueness and regularity of solutions you should check this book as a primary reference, and let me give you two reasons for that.1. The statements of the theorems are general enough to fit into a large number of nonlinear PDE problem such as elliptic, parabolic or hyperbolic equations. In fact, many of M. Taylor’s theorems have two version, one in a euclidean space and another on a manifold.2. The Fixed point methods in Banach spaces. This is my favorite part because Taylor shows in a very clear and emphatic manner why Fixed point theorems are important and how to use them properly.Just a warning: the level of this book is for graduate students in mathematics.For a more basic (or even first) approach to PDEs my call is always Evans’ book

⭐. For a second opinion on nonlinear equations I strongly recomend Smoller’s book

⭐.

⭐One of the most important and subtle features of mathematics is the great difference existing between linearity and non-linearity. One always tries to linearize whenever possible, because linear problems are easier to solve, but unfortunately the world is not linear, and we have to learn how to deal with non-linear problems.This volume discusses thoroughly some very important cases of non-linear PDE’s which are important in mathematical physics or that possess intrinsic theoretical interest.The contents are: Function space and operator theory for nonlinear analysis; nonlinear elliptic equations; nonlinear parabolic equations; nonlinear hyperbolic equations; Euler and Navier-Stokes equations for incompressible fluids; Einstein’s equations.A little bit more advanced and speciallized than the other two volumes, but still useful for a broad community. A graduate student with a solid background will find it perfectly acquaintable. Includes lots of excercises and references.Please read my other reviews (just click on my name above).

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