Potential Theory pp 107-147 | Cite as

# Green Functions

## 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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