Partial differential equations. I: Basic theory. 2nd ed.

*(English)*Zbl 1206.35002
Applied Mathematical Sciences 115. New York, NY: Springer (ISBN 978-1-4419-7054-1/hbk; 978-1-4419-7055-8/ebook). xxii, 654 p. (2011).

This is the first of three volumes on partial differential equations, this one introduces basic examples arising in continuum mechanics, electromagnetism, complex analysis and other areas, and develops a number of tools for their solution, in particular Fourier analysis, distribution theory, and Sobolev spaces. These tools are then applied to the treatment of basic problems in linear PDEs, including the Laplace equation, heat equation, and wave equation, as well as more general elliptic, parabolic, and hyperbolic equations. In this first volume there are six chapters and Appendix A and Appendix B.

Chapter 1 is devoted to basic theory of ODEs and vector fields.

Chapter 2 analyses the Laplace equation and wave equation.

Chapter 3 concerns Fourier analysis, distributions and constant-coefficient linear PDE.

Chapter 4 is focused on Sobolev spaces.

Chapter 5 is devoted to linear elliptic equations.

Chapter 6 concerns linear evolution equations.

In Appendix A, the author gives an outline of functional analysis.

In Appendix B, manifolds, vector bundles and Lie groups are analysed.

The book is targeted at graduate students 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, the space of pseudodifferential operators of harmonic oscillator type, and an index formula for elliptic systems of such operators. 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.

Chapter 1 is devoted to basic theory of ODEs and vector fields.

Chapter 2 analyses the Laplace equation and wave equation.

Chapter 3 concerns Fourier analysis, distributions and constant-coefficient linear PDE.

Chapter 4 is focused on Sobolev spaces.

Chapter 5 is devoted to linear elliptic equations.

Chapter 6 concerns linear evolution equations.

In Appendix A, the author gives an outline of functional analysis.

In Appendix B, manifolds, vector bundles and Lie groups are analysed.

The book is targeted at graduate students 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, the space of pseudodifferential operators of harmonic oscillator type, and an index formula for elliptic systems of such operators. 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.

Reviewer: Vincenzo Vespri (Firenze)

##### MSC:

35-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to partial differential equations |

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |

35K05 | Heat equation |

58Jxx | Partial differential equations on manifolds; differential operators |