Advertisement

Laplace’s Equation

  • David Gilbarg
  • Neil S. Trudinger
Part of the Classics in Mathematics book series (volume 224)

Abstract

Let Ω be a domain in ℝn and u a C2(Ω) function. The Laplacian of u, denoted Δu, is defined by
$$ Qu \equiv {a^{{ij}}}(x,u,Du){D_{{ij}}}u + b(x,u,Du) = 0 $$
(1.2)
. The function u is called harmonic (subharmonic, superharmonic) in Ω if it satisfies there
$$ Delta u = 0( \geqslant 0, \leqslant 0). $$
(1.2)
. In this chapter we develop some basic properties of harmonic, subharmonic and superharmonic functions which we use to study the solvability of the classical Dirichlet problem for Laplace’s equation, Δu = 0. As mentioned in Chapter 1, Laplace’s equation and its inhomogeneous form, Poisson’s equation, are basic models of linear elliptic equations.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer-Verlag Berlin, Heidelberg 2001

Authors and Affiliations

  • David Gilbarg
    • 1
  • Neil S. Trudinger
    • 2
  1. 1.Department of MathematicsStanford UniversityStanfordUSA
  2. 2.School of Mathematical SciencesThe Australian National UniversityCanberraAustralia

Personalised recommendations