This book is intended for students who wish to get an introduction to the theory of partial differential equations. The author focuses on elliptic equations and systematically develops the relevant existence schemes, always with a view towards nonlinear problems. These are maximum principle methods (particularly important for numerical analysis schemes), parabolic equations, variational methods, and continuity methods. This book 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. Connections between elliptic, parabolic, and hyperbolic equations are explored, as well as the connection with Brownian motion and semigroups. This book can be utilized for a one-year course on partial differential equations. Jurgen Jost is Director of the Max Planck Institute for Mathematics in the Sciences and Professor of Mathematics at the University of Leipzig. He is the author of a number of Springer books, including Postmodern Analysis (1998), Compact Riemann Surfaces (1997) and Riemannian Geometry and Geometric Analysis (1995).
The present book is an expanded translation of the original German version, Partielle Differentialgleichungen (1998).
Table of Contents:
Introduction.- The Laplace equation as the prototype of an elliptic partial differential equation of 2nd order.- The maximum principle.- Existence techniques I: methods based on the maximum principle.- Existence techniques II: Parabolic methods. The Head equation.- The wave equation and its connections with the Laplace and heat equation.- The heat equation, semigroups, and Brownian motion.- The Dirichlet principle. Variational methods for the solution of PDE (Existence techniques III).- Sobolev spaces and L2 regularity theory.- Strong solutions.- The regularity theory of Schauder and the continuity method (Existence techniques IV).- The Moser iteration method and the reqularity theorem of de Giorgi and Nash.- Banach and Hilbert spaces. The Lp-spaces.- Bibliography.
Review :
From the reviews: MATHEMATICAL REVIEWS
.,"the composition of this book is somewhat classical. Indeed the author covers the main properties of the elliptic, parabolic and hyperbolic equations. But he always adds some interesting extensions or links between these chaptersa ]this textbook is self-containeda ]Because of the nice global presentation, I recommend this book to students and young researchers who need the now classical properties of these second-order partial differential equations. Teachers will also find in this textbook the basis of an introductory course on second-order partial differential equations."
"The author covers the main properties of the elliptic parabolic and hyperbolic equations. But he always adds some interesting extensions or links between these chapters. a ] With an appendix on general results concerning Banach or Hilbert spaces, this textbook is self-contained. a ] Because of the nice global presentation, I recommend this book to students and young researchers who need the now classical properties of these second-order partial differential equations." (Alain Brillard, Mathematical Reviews, 2003f)
"Josta (TM)s book a ] focuses mainly on elliptic PDEs and gives a comprehensive overview of the modern theory of solving such equations. a ] There are extremely helpful summaries and a handful of exercises at the end of each of the 11 chapters; an appendix covers background functional analysis. a ] Throughout, Jost achieves an impeccable blend of motivation, orientation and analysis a ] . Beautifully written and superbly well-organised, I strongly recommend this book to anyone seeking a stylish, balanced, up-to-date survey of this central area ofmathematics." (Nick Lord, The Mathematical Gazette, Vol. 88 (512), 2004)
From the reviews: MATHEMATICAL REVIEWS
.."the composition of this book is somewhat classical. Indeed the author covers the main properties of the elliptic, parabolic and hyperbolic equations. But he always adds some interesting extensions or links between these chaptersthis textbook is self-containedBecause of the nice global presentation, I recommend this book to students and young researchers who need the now classical properties of these second-order partial differential equations. Teachers will also find in this textbook the basis of an introductory course on second-order partial differential equations."
"The author covers the main properties of the elliptic parabolic and hyperbolic equations. But he always adds some interesting extensions or links between these chapters. With an appendix on general results concerning Banach or Hilbert spaces, this textbook is self-contained. Because of the nice global presentation, I recommend this book to students and young researchers who need the now classical properties of these second-order partial differential equations." (Alain Brillard, Mathematical Reviews, 2003f)
"Josts book focuses mainly on elliptic PDEs and gives a comprehensive overview of the modern theory of solving such equations. There are extremely helpful summaries and a handful of exercises at the end of each of the 11 chapters; an appendix covers background functional analysis. Throughout, Jost achieves an impeccable blend of motivation, orientation and analysis . Beautifully written and superbly well-organised, I strongly recommend this book to anyone seeking a stylish, balanced, up-to-date survey of this central area of mathematics." (Nick Lord, The MathematicalGazette, Vol. 88 (512), 2004)
