Skip to main content

Dominance of Positivity in Linear Equations

  • 1485 Accesses

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

Abstract

In Section 1.2 we mentioned that although the Green functionG Δ2 Ω for the clamped plate boundary value problem

$$ \left\{\begin{array}{l}\Delta^2 u = f\ in\ \Omega \\ u\ =\ |\nabla u|\ =\ 0\ on\ \partial\ \Omega \end{array}\right.\ $$

is in general sign changing, it is very hard to display its negative part in numerical simulations or in real world experiments. Moreover, numerical work in nonlinear elliptic fourth order equations suggests that maximum or comparison principles are violated only to a “small extent”. Nevertheless, we do not yet have tools at hand to give this feeling a precise form and, in particular, a quantitative form which might prove to be useful also for nonlinear higher order equations.

Keywords

  • Green Function
  • Dirichlet Boundary Condition
  • Smooth Domain
  • Biharmonic Operator
  • Order Elliptic Operator

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   59.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   79.99
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Filippo Gazzola .

Rights and permissions

Reprints and Permissions

Copyright information

© 2010 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Gazzola, F., Grunau, HC., Sweers, G. (2010). Dominance of Positivity in Linear Equations. In: Polyharmonic Boundary Value Problems. Lecture Notes in Mathematics(), vol 1991. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-12245-3_6

Download citation