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 well as the connection with Brownian motion and semigroups. This second edition features a new chapter on reaction-diffusion equations and systems.
Author(s): Jürgen Jost (auth.)
Series: Graduate Texts in Mathematics 214
Edition: 2nd
Publisher: Springer New York
Year: 2002
Language: English
Pages: 373
Tags: Partial Differential Equations; Mathematical and Computational Physics; Numerical and Computational Methods
Introduction: What Are Partial Differential Equations?....Pages 1-6
The Laplace Equation as the Prototype of an Elliptic Partial Differential Equation of Second Order....Pages 7-30
The Maximum Principle....Pages 31-50
Existence Techniques I: Methods Based on the Maximum Principle....Pages 51-75
Existence Techniques II: Parabolic Methods. The Heat Equation....Pages 77-112
The Wave Equation and Its Connections with the Laplace and Heat Equations....Pages 113-125
The Heat Equation, Semigroups, and Brownian Motion....Pages 127-156
The Dirichlet Principle. Variational Methods for the Solution of PDEs (Existence Techniques III)....Pages 157-192
Sobolev Spaces and L 2 Regularity Theory....Pages 193-242
Strong Solutions....Pages 243-254
The Regularity Theory of Schauder and the Continuity Method (Existence Techniques IV)....Pages 255-274
The Moser Iteration Method and the Regularity Theorem of de Giorgi and Nash....Pages 275-307
