# Harmonic space

A topological space with a (linear) sheaf of continuous real-valued functions in which three fundamental properties of classical harmonic functions (cf. Harmonic function) are axiomatically fixed. These are: convergence as expressed by the second Harnack theorem; the maximum/minimum principle; and the solvability of the Dirichlet problem for a sufficiently broad class of open sets in . The functions in are called harmonic functions. The advantage of this axiomatic approach consists in the fact that if it is adopted, the theory comprises not only solutions of the Laplace equation, but also of certain other equations of elliptic and parabolic type. Let be a locally compact topological space. A sheaf of functions on is understood to be a mapping defined on the family of all open sets of such that: 1) is a family of functions on ; 2) if , then the restriction of any function in to belongs to ; and 3) for any family , a function on belongs to if for all its restriction to belongs to . A sheaf of functions is called hyperharmonic if, for any , is a convex cone of lower semi-continuous finite numerical functions on . A sheaf of functions is said to be harmonic if, for any , is a real vector space of continuous functions on ; the harmonic sheaf

is used in what follows.

A locally compact space is called a harmonic space if the following axioms are satisfied [3].

The positivity axiom: is non-degenerate at all points , i.e. for any there exists a function defined in a neighbourhood of such that .

The convergence axiom: If an increasing sequence of functions in is locally bounded, then it converges towards a function in .

The resolutivity axiom: There exists a basis of resolutive open sets , i.e. sets such that for any continuous function of compact support on there exists a solution of the Dirichlet problem for from , understood in the generalized sense of Wiener–Perron (cf. Perron method).

The axiom of completeness: If a lower semi-continuous function of on , which is lower finite, satisfies the condition

on , for any relatively compact set with , then .

The Euclidean space , , with the sheaf of classical solutions of the Laplace equation or of the thermal-conductance equation (heat equation) forms a harmonic space. There are several other variants of the axiomatics of harmonic spaces. Harmonic spaces are locally connected and do not contain isolated points; they have a basis of connected resolutive sets.

A hyperharmonic function on a harmonic space is called superharmonic if for any relatively compact resolutive set the function is harmonic on . A positive superharmonic function for which any positive harmonic minorant is identically equal to zero is called a potential. A harmonic space is called -harmonic (-harmonic) if for any there exists a positive superharmonic function (or, respectively, a potential ) on such that .

Any harmonic space can be covered by open sets which satisfy the minimum principle in the following form: If a hyperharmonic function is positive outside the intersection of with any compact set in and if

for all , then . In the case of a -harmonic space this minimum principle is satisfied for all open sets. The Euclidean space with the sheaf of classical solutions of the Laplace equation is an -harmonic space if , and is a -harmonic space if and only if ; the space , , with the sheaf of solutions of the heat equation is a -harmonic space.

The principal problems of the theory of harmonic spaces include the theory of solvability of the Dirichlet problem, including the behaviour of the generalized solution of this problem at boundary points. The theory of the capacity of a set in a harmonic space, problems of balayage (cf. Balayage method) and the Robin problem have been studied.

#### References

 [1] M. Brélot, "Lectures on potential theory" , Tata Inst. (1960) [2] H. Bauer, "Harmonische Räume und ihre Potentialtheorie" , Springer (1966) [3] C. Constantinescu, A. Cornea, "Potential theory on harmonic spaces" , Springer (1972) [4] M. Brelot, "On topologies and boundaries in potential theory" , Springer (1971)