Partial Differential Equations (eBook)

Jürgen Jost (Autor)

eBook Download: PDF

2012 | 3rd ed. 2013
XIV, 410 Seiten
Springer New York (Verlag)
978-1-4614-4809-9 (ISBN)

Infos

This extensively revised and updated new edition of a core text on partial differential equations features clear and systematic discussion of the relations between different types of PDEs. It covers new Hamack-type techniques, Schauder estimates, and more.

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This book offers an ideal graduate-level introduction to the theory of partial differential equations.  The first part of the book describes the basic mathematical problems and structures associated with elliptic, parabolic, and hyperbolic partial differential equations, and explores the connections between these fundamental types. Aspects of  Brownian motion or pattern formation processes are also presented. The second part focuses on existence schemes and develops estimates for solutions of elliptic equations, such as Sobolev space theory, weak and strong solutions, Schauder estimates, and Moser iteration. In particular, the reader will learn the basic techniques underlying current research in elliptic partial differential equations.This revised and expanded third edition is enhanced with many additional examples that will help motivate the reader.  New features include a reorganized and extended chapter on hyperbolic equations, as well as a new chapter on the relations between different types of partial differential equations, including first-order hyperbolic systems, Langevin and Fokker-Planck equations, viscosity solutions for elliptic PDEs, and much more. Also, the new edition contains additional material on systems of elliptic partial differential equations, and it explains in more detail how the Harnack inequality can be used for the regularity of solutions.

Autor

Jürgen Jost is currently a codirector of the Max Planck Institute for Mathematics in the Sciences and an honorary professor of mathematics at the University of Leipzig.

Inhalt

Preface.- Introduction: What are Partial Differential Equations?.- 1 The Laplace equation as the Prototype of an Elliptic Partial Differential Equation of Second Order.- 2 The Maximum Principle.- 3 Existence Techniques I: Methods Based on the Maximum Principle.- 4 Existence Techniques II: Parabolic Methods. The Heat Equation.- 5 Reaction-Diffusion Equations and Systems.- 6 Hyperbolic Equations.- 7 The Heat Equation, Semigroups, and Brownian Motion.- 8 Relationships between Different Partial Differential Equations.- 9 The Dirichlet Principle. Variational Methods for the Solutions of PDEs (Existence Techniques III).- 10 Sobolev Spaces and L^2 Regularity theory.- 11 Strong solutions.- 12 The Regularity Theory of Schauder and the Continuity Method (Existence Techniques IV).- 13The Moser Iteration Method and the Regularity Theorem of de Giorgi and Nash.- Appendix: Banach and Hilbert spaces. The L^p-Spaces.- References.- Index of Notation.- Index.

Produkt-Details

• Erscheint lt. Verlag: 1.11.2013
• Reihe/Serie: Graduate Texts in Mathematics
• Verlagsort: New York
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Naturwissenschaften ► Physik / Astronomie
• Themenwelt: Technik
• Schlagworte: Brownian motion • eigenvalues • Harmonic Functions • Harnack inequality • heat equation • Hilbert space methods • Maximum principle • Moser iteration • Nonlinear Partial Differential Equations • pattern formation • reaction-diffusion equations and systems • Schauder estimates • semigroups • Sobolev spaces • Turing mechanism • wave equation
• ISBN-10: 1-4614-4809-3 / 1461448093
• ISBN-13: 978-1-4614-4809-9 / 9781461448099