Harmonic space

A topological space $ X $ with a (linear) sheaf $ \mathfrak H $ 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 $ X $. The functions in $ \mathfrak H $ 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 $ X $ be a locally compact topological space. A sheaf of functions on $ X $ is understood to be a mapping $ \mathfrak F $ defined on the family of all open sets $ U, V \dots $ of $ X $ such that: 1) $ \mathfrak F ( U) $ is a family of functions on $ U $; 2) if $ U \subset V $, then the restriction of any function in $ \mathfrak F ( V) $ to $ U $ belongs to $ \mathfrak F ( U) $; and 3) for any family $ \{ U _ {i} \} _ {i \in I } $, a function on $ \cup _ {i \in I } U _ {i} $ belongs to $ \mathfrak F ( \cup _ {i \in I } U ) $ if for all $ i \in I $ its restriction to $ U _ {i} $ belongs to $ \mathfrak F ( U _ {i} ) $. A sheaf of functions $ \mathfrak U $ is called hyperharmonic if, for any $ U $, $ \mathfrak U ( U) $ is a convex cone of lower semi-continuous finite numerical functions on $ U $. A sheaf of functions $ \mathfrak H $ is said to be harmonic if, for any $ U $, $ \mathfrak H ( U) $ is a real vector space of continuous functions on $ U $; the harmonic sheaf

$$ \mathfrak H : U \rightarrow \mathfrak U ( U) \cap (- \mathfrak U ( U)) $$

is used in what follows.

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

The positivity axiom: $ \mathfrak H $ is non-degenerate at all points $ x \in X $, i.e. for any $ x \in X $ there exists a function $ u \in \mathfrak H $ defined in a neighbourhood of $ x $ such that $ u( x) \neq 0 $.

The convergence axiom: If an increasing sequence of functions in $ \mathfrak H ( U) $ is locally bounded, then it converges towards a function in $ \mathfrak H ( U) $.

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

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

$$ \sup \{ H ( U , f ) : u \geq f \in C ( \partial V ) \} = \ \mu ^ {V} u \leq u $$

on $ V $, for any relatively compact set $ V $ with $ \overline{V}\; \subset U $, then $ u \in \mathfrak U ( U) $.

The Euclidean space $ \mathbf R ^ {n} $, $ n \geq 2 $, 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 $ u $ on a harmonic space $ X $ is called superharmonic if for any relatively compact resolutive set $ V $ the function $ \mu ^ {V} u $ is harmonic on $ V $. A positive superharmonic function for which any positive harmonic minorant is identically equal to zero is called a potential. A harmonic space $ X $ is called $ \mathfrak s $- harmonic ( $ \mathfrak p $- harmonic) if for any $ x \in X $ there exists a positive superharmonic function $ u $( or, respectively, a potential $ u $) on $ X $ such that $ u( x) > 0 $.

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

$$ \lim\limits _ {x \rightarrow y } \ \inf u ( x) \geq 0 $$

for all $ y \in \partial U $, then $ u \geq 0 $. In the case of a $ \mathfrak p $- harmonic space this minimum principle is satisfied for all open sets. The Euclidean space $ \mathbf R ^ {n} $ with the sheaf of classical solutions of the Laplace equation is an $ \mathfrak s $- harmonic space if $ n \geq 1 $, and is a $ \mathfrak p $- harmonic space if and only if $ n \geq 3 $; the space $ \mathbf R ^ {n} \times \mathbf R ^ {1} $, $ n \geq 1 $, with the sheaf of solutions of the heat equation is a $ \mathfrak p $- 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)

Comments

Any $ \mathfrak p $- harmonic space $ X $ with a countable base and for which the function 1 is superharmonic admits the construction of a suitable Markov process, such that potential-theoretic notions of $ X $ correspond to potential-theoretic notions of the process.

See also Potential theory; Potential theory, abstract.

References

[a1]	H. Bauer, "Harmonische Räume" , Jahrbuch Ueberblicke Mathematik , B.I. Wissenschaftsverlag Mannheim (1981) pp. 9–35
[a2]	M. Brelot (ed.) H. Bauer (ed.) J.-M. Bony (ed.) J. Deny (ed.) G. Mokobodzki (ed.) , Potential theory (CIME, Stresa, 1969) , Cremonese (1970)