Progress in Nonlinear Differential Equations and Their Applications

Implicit Partial Differential Equations

Authors: Dacorogna, Bernard, Marcellini, Paolo

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About this book

Nonlinear partial differential equations has become one of the main tools of mod­ ern mathematical analysis; in spite of seemingly contradictory terminology, the subject of nonlinear differential equations finds its origins in the theory of linear differential equations, and a large part of functional analysis derived its inspiration from the study of linear pdes. In recent years, several mathematicians have investigated nonlinear equations, particularly those of the second order, both linear and nonlinear and either in divergence or nondivergence form. Quasilinear and fully nonlinear differential equations are relevant classes of such equations and have been widely examined in the mathematical literature. In this work we present a new family of differential equations called "implicit partial differential equations", described in detail in the introduction (c.f. Chapter 1). It is a class of nonlinear equations that does not include the family of fully nonlinear elliptic pdes. We present a new functional analytic method based on the Baire category theorem for handling the existence of almost everywhere solutions of these implicit equations. The results have been obtained for the most part in recent years and have important applications to the calculus of variations, nonlin­ ear elasticity, problems of phase transitions and optimal design; some results have not been published elsewhere.

Reviews

"Provides a self-contained development of the new kind of differential equations… Includes many examples helpful in understanding the theory and is well [and] clearly written."

—Mathematical Reviews

"…Topics of the book have important applications in variational calculus, nonlinear elasticity, phase transitions, and optimal design. In particular, Part IV (Chapters 7-9) includes the singular value case, the case of potential wells, and the complex eikonal equation. A lot of background material, as viscosity solutions, quasiconvexity and related topics, piecewise polynomial approximation in Sobolev spaces, convex integration, or nonclassical Vitali-type covering theorems, is presented in the book, which makes it essentially self-contained. Most of the results are original from the authors, some results are published even first time. This makes the book a unique introduction to this new topic, and will be found useful by experts in nonlinear analysis in general and PDEs in particular, as well as by advanced graduate students in this field."

–Zentralblatt Math


Table of contents (10 chapters)

  • Introduction

    Dacorogna, Bernard (et al.)

    Pages 1-30

  • First Order Equations

    Dacorogna, Bernard (et al.)

    Pages 33-68

  • Second Order Equations

    Dacorogna, Bernard (et al.)

    Pages 69-93

  • Comparison with Viscosity Solutions

    Dacorogna, Bernard (et al.)

    Pages 95-117

  • Some Preliminary Results

    Dacorogna, Bernard (et al.)

    Pages 121-140

Buy this book

eBook $99.00
price for USA (gross)
  • ISBN 978-1-4612-1562-2
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $169.00
price for USA
  • ISBN 978-0-8176-4121-4
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Softcover $129.00
price for USA
  • ISBN 978-1-4612-7193-2
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Implicit Partial Differential Equations
Authors
Series Title
Progress in Nonlinear Differential Equations and Their Applications
Series Volume
37
Copyright
1999
Publisher
Birkhäuser Basel
Copyright Holder
Birkhäuser Boston
eBook ISBN
978-1-4612-1562-2
DOI
10.1007/978-1-4612-1562-2
Hardcover ISBN
978-0-8176-4121-4
Softcover ISBN
978-1-4612-7193-2
Series ISSN
1421-1750
Edition Number
1
Number of Pages
XIII, 273
Topics