Lecture Notes in Mathematics
cover

Polyharmonic Boundary Value Problems

Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains

Authors: Gazzola, Filippo, Grunau, Hans-Christoph, Sweers, Guido

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About this book

This monograph covers higher order linear and nonlinear elliptic boundary value problems in bounded domains, mainly with the biharmonic or poly-harmonic operator as leading principal part. Underlying models and, in particular, the role of different boundary conditions are explained in detail. As for linear problems, after a brief summary of the existence theory and Lp and Schauder estimates, the focus is on positivity or - since, in contrast to second order equations, a general form of a comparison principle does not exist - on “near positivity.” The required kernel estimates are also presented in detail. As for nonlinear problems, several techniques well-known from second order equations cannot be utilized and have to be replaced by new and different methods. Subcritical, critical and supercritical nonlinearities are discussed and various existence and nonexistence results are proved. The interplay with the positivity topic from the first part is emphasized and, moreover, a far-reaching Gidas-Ni-Nirenberg-type symmetry result is included. Finally, some recent progress on the Dirichlet problem for Willmore surfaces under symmetry assumptions is discussed.

Reviews

From the reviews:

“This is an excellent book, full of well-explained ideas and techniques on the subject, and can be used as a textbook in an advanced course dealing with higher-order elliptic problems. The proofs of almost all of the theorems directly related to the higher-order elliptic problems are complete and well written. In general, the book itself is written in a very clear, pleasurable style, including a wealth of useful diagrams and figures, some of them in color.” (Rodney Josué Biezuner, Mathematical Reviews, Issue 2011 h)

“The main tasks of the present Lecture Notes in Mathematics volume are nonlinear problems and positivity statements for higher order elliptic equations involving polyharmonic operators. In particular the biharmonic operator and semilinear operators related to it are investigated. … That the authors are experienced researchers on the topic of the volume becomes evident from the excellent, clear and well motivated presentation. … Whoever is interested in an exciting subject of the modern theory of higher order elliptic equations is recommended to study this exposition.” (Heinrich Begehr, Zentralblatt MATH, Vol. 1239, 2012)


Table of contents (8 chapters)

Table of contents (8 chapters)
  • Models of Higher Order

    Pages 1-25

    Gazzola, Filippo (et al.)

  • Linear Problems

    Pages 27-60

    Gazzola, Filippo (et al.)

  • Eigenvalue Problems

    Pages 61-98

    Gazzola, Filippo (et al.)

  • Kernel Estimates

    Pages 99-146

    Gazzola, Filippo (et al.)

  • Positivity and Lower Order Perturbations

    Pages 147-185

    Gazzola, Filippo (et al.)

Buy this book

eBook $59.99
price for USA (gross)
  • ISBN 978-3-642-12245-3
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $79.99
price for USA
  • ISBN 978-3-642-12244-6
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.

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Bibliographic Information

Bibliographic Information
Book Title
Polyharmonic Boundary Value Problems
Book Subtitle
Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains
Authors
Series Title
Lecture Notes in Mathematics
Series Volume
1991
Copyright
2010
Publisher
Springer-Verlag Berlin Heidelberg
Copyright Holder
Springer-Verlag Berlin Heidelberg
eBook ISBN
978-3-642-12245-3
DOI
10.1007/978-3-642-12245-3
Softcover ISBN
978-3-642-12244-6
Series ISSN
0075-8434
Edition Number
1
Number of Pages
XVIII, 423
Number of Illustrations
18 b/w illustrations
Topics