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Green Functions

  • Lester L. Helms
Part of the Universitext book series (UTX)

Abstract

In 1828, Green introduced a function, which he called a potential, for calculating the distribution of a charge on a surface bounding a region in R n in the presence of external electromagnetic forces. The argument that led to the function G B introduced in Section 1.5 is precisely the argument made by Green in [26]. The function introduced by Green is now called the Green function. After Green, the most general existence theorem for Green functions was proven by Osgood in 1900 for simply connected regions in the plane (see [49]). In this chapter it will be shown that a Green function can be defined for some sets, but not all, which serve the same purpose as G B . The sets for which this is true are called Greenian sets. This chapter will culminate in an important theorem of F. Riesz which states that a nonnegative superharmonic on a Greenian set has a decomposition into the sum of a Green potential and a harmonic function.

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Copyright information

© Springer-Verlag London Limited 2009

Authors and Affiliations

  • Lester L. Helms
    • 1
  1. 1.Department of MathematicsUniversity of IllinoisUrbanaUSA

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