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Green Functions

  • Lester L. HelmsEmail author
Chapter
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Part of the Universitext book series (UTX)

Abstract

The Green function is a function devised by George Green in 1828 to construct solutions of Poisson’s equation \(\triangle u = f\) on a domain in \(R^n\). Following a formal definition of the Green function, it is shown that every open subset of \(R^n, n \ge 3\), has a Green function whereas \(R^2\) does not have a Green function nor do domains in \(R^2\) whose complements are “too small”. A classical method of constructing Green functions, known as “the method of images”, is illustrated by way of exercises. The potential energy of a unit mass at a point of \(R^n\) due to a mass distributed in a region of \(R^n\) is an example of a Green potential of a measure. This concept is extended to regions having a Green functions and to signed measures on the region. Properties of Green potentials are used to characterize superharmonic functions and to construct a potential on a region known as the “Lebesgue Spine” to show that there are regions for which the solution of the Dirichlet problem for the region does not take on the value of the boundary function at the cusp of the spine.

Keywords

Green Function Nonnegative Superharmonic Function George Green Greatest Harmonic Minorant Measurable Boundary Function 
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.

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Copyright information

© Springer-Verlag London 2014

Authors and Affiliations

  1. 1.University of IllinoisUrbanaUSA

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