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Polyharmonic Boundary Value Problems

Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains

  • Filippo Gazzola
  • Hans-Christoph Grunau
  • Guido Sweers

Part of the Lecture Notes in Mathematics book series (LNM, volume 1991)

Table of contents

  1. Front Matter
    Pages i-xviii
  2. Filippo Gazzola, Hans-Christoph Grunau, Guido Sweers
    Pages 1-25
  3. Filippo Gazzola, Hans-Christoph Grunau, Guido Sweers
    Pages 27-60
  4. Filippo Gazzola, Hans-Christoph Grunau, Guido Sweers
    Pages 61-98
  5. Filippo Gazzola, Hans-Christoph Grunau, Guido Sweers
    Pages 99-146
  6. Filippo Gazzola, Hans-Christoph Grunau, Guido Sweers
    Pages 147-185
  7. Filippo Gazzola, Hans-Christoph Grunau, Guido Sweers
    Pages 187-226
  8. Filippo Gazzola, Hans-Christoph Grunau, Guido Sweers
    Pages 227-370
  9. Filippo Gazzola, Hans-Christoph Grunau, Guido Sweers
    Pages 371-392
  10. Back Matter
    Pages 393-429

About this book

Introduction

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.

Keywords

Almost positivity Higher order elliptic boundary value problems equation function polyharmonic Gidas-Ni-Nirenberg-type symmetry result polyharmonic Green function estimates

Authors and affiliations

  • Filippo Gazzola
    • 1
  • Hans-Christoph Grunau
    • 2
  • Guido Sweers
    • 3
  1. 1., Dipartimento di MatematicaPolitecnico MilanoMilanoItaly
  2. 2., Institut f. Analysis und NumerikOtto von Guericke-UniversitätMagdeburgGermany
  3. 3., Mathematisches InstitutUniversität zu KölnKölnGermany

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-642-12245-3
  • Copyright Information Springer-Verlag Berlin Heidelberg 2010
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-642-12244-6
  • Online ISBN 978-3-642-12245-3
  • Series Print ISSN 0075-8434
  • Series Online ISSN 1617-9692
  • Buy this book on publisher's site