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Green space

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A topological space $X$ on which harmonic and superharmonic functions (cf. Harmonic function; Subharmonic function) are defined and for which a Green function exists (for the Dirichlet problem in the class of harmonic functions) or, which amounts to the same thing, for which there exists a non-constant superharmonic function. More exactly, let $X$ be an $E$-space, i.e. a connected separable topological space in which: 1) each point $x\in X$ has an open neighbourhood $V_x$ homeomorphic to some open set $V_x'$ of a Euclidean space $\mathbf R^n$ (or of its Aleksandrov compactification $\overline{\mathbf R^n}$); and 2) the images of any non-empty intersection $V_x\cap V_y$ (under the two homeomorphisms to $\mathbf R^n$ or $\overline{\mathbf R^n}$) of two neighbourhoods in $V_x'$ and $V_y'$ are isometric, and are conformally equivalent if $n=2$. Harmonic and superharmonic functions on an $E$-space $X$ are locally defined by passing to the images $V_x'$. If, in addition, there exists a non-constant positive superharmonic function on $X$ or, which amounts to the same thing, a positive potential, $X$ is known as a Green space. Thus, the Euclidean space $\mathbf R^n$, its compactification $\overline{\mathbf R^n}$ ($n\geq2$) and Riemann surfaces are all $E$-spaces. Here, $\mathbf R^n$ $(n\geq3)$ and Riemann surfaces of hyperbolic type (cf. Riemann surfaces, classification of) are Green spaces, while $\mathbf R^2$ and Riemann surfaces of parabolic type are not. Any domain in a Green space $X$ is again a Green space.

A harmonic space $X$ with a positive potential on it can also be regarded as a generalization of a Green space in the framework of axiomatic potential theory (cf. Potential theory, abstract).

References

[1] M. Brelot, "On topologies and boundaries in potential theory" , Springer (1971)
[2] M. Brélot, G. Choquet, "Espaces et lignes de Green" Ann. Inst. Fourier , 3 (1952) pp. 199–263
How to Cite This Entry:
Green space. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Green_space&oldid=34097
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article