Partial Differential Equations (Graduate Texts in Mathematics)
October 5, 2010 by admin
Filed under Mathematical Physics
Partial Differential Equations (Graduate Texts in Mathematics)
This book offers an ideal introduction to the theory of partial differential equations. It focuses on elliptic equations and systematically develops the relevant existence schemes, always with a view towards nonlinear problems. It also develops the main methods for obtaining estimates for solutions of elliptic equations: Sobolev space theory, weak and strong solutions, Schauder estimates, and Moser iteration. It also explores connections between elliptic, parabolic, and hyperbolic equations as w
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(out of 2 reviews)
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Review by A. Castro for Partial Differential Equations (Graduate Texts in Mathematics)
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I’m no specialist in PDE’s but I boldly would like to review the present book.
This book is intended(in my personal view) as a in-depth-but-not-pedantic introduction to Elliptic equations (which, if one considers the original title in german “Partielle Differentialgleichungen – Elliptische Gleichungen” makes complete sense).
As any other descent book, it doesn’t aim at list down results in a encyclopedic way, but rather tour-guide the reader in the beautiful subject of Elliptic PDE’s.
Motivated by Dirichlet’s principle, the author introduces variational methods for Elliptic PDE’s and a good deal of the regularity program for that class of equations(including non-linear equations). Most of what follows the first chapter, is a successful attempt to generalize to other elliptic equations the techniques useful in the qualitative theory of the Dirichlet problem for the Poisson equation. Some of the methods, as far as I understand them, can be generalized to Schroedinger’s equation too.
The interested reader can consult also the book on analysis by Elliot Lieb and the textbook on variational methods by Prof. S. Hildebrandt(and references therein).
Review by areader for Partial Differential Equations (Graduate Texts in Mathematics)
Rating:
This is an excellent book for a PDE course with a strong vias toward
elliptic and parabolic equations. Its lack of information related to hyperbolic equations is its weakest point. However, the author is an
excellent writer and it is pleasure to read this book.